Issue 76

Beam Dynamics Newsletter

3.11 Simulation of an FFA Return Channel ERL in a RHIC-based Electron-Ion Collider

François Méot, Brookhaven National Laboratory C-AD, Upton, NY USA.

Abstract

The linac-ring electron-ion collider designed at BNL is based on a 20 GeV ERL comprised of a 1.3 GeV linac and two 3.8 km FFA lattice return loops placed along the RHIC heavy ion collider. This document introduces the simulation of the FFA ERL in Zgoubi, including a start-to-end 6-D simulation of the acceleration-deceleration cycle of a polarized electron bunch.

Introduction

The design of eRHIC linac-ring EIC ERL  [1] is based on a 1.322 GeV linac, and on two FFA loops located alongside RHIC, Fig. 1. The low energy loop recirculates the electron bunches four times on the way up and four times on the way down (\( 0.012\! \stackrel {up ~ }{\rightarrow } \! 5.300\! \stackrel {down}{\rightarrow }\!0.012 \), step 1.322 GeV). The second loop is of concern in this document; it recirculates the electron bunches 12 times up and 11 times down,

\[ 5.3 \stackrel {up ~ }{\longrightarrow } 21.164 \stackrel {down}{\longrightarrow }\ 5.3\,\mathrm
{GeV} \]

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Figure 1: eRHIC ERL with its two recirculation loops alongside RHIC. The top left box shows a cross-section of the low and high energy multiple-beam FFA channels. The 1.322 GeV linac is located in RHIC IR2, it is connected to the FFA loops by a merger section (resp. spreader) at its upstream (resp. downstream) end.

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Figure 2: Transverse excursion of the 12 periodic orbits (for 12 different energies) across the FFA cell magnets, shown in the respective magnet frames (\( x=0 \) is the quadrupole axis). The optical axes of the quadrupoles in the arc cell are radially shifted by 13.48 mm with respect to one another; this ensures 8.73 mrad orbit bending across the cell.

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Figure 3: Magnetic field along the 12 orbits in a hard-edge model.

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Figure 4: Cell tunes and chromaticities versus energy; the vertical bars correspond to the 12 design energies. Features of the linear FFA cell include tunes decreasing with energy since quadrupole gradients are fixed, and natural chromaticity decreasing with energy.

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Figure 5: Optical functions at 6.622 GeV (left) and 21.164 GeV (right) from stepwise ray-tracing across the FFA cell.

In the following, §3.11.1 discusses the general properties of the 3.8 km FFA loop. The ERL lattice is described in §3.11.2, A 23-pass acceleration-deceleration cycle of a 6D bunch is then simulated. The present document is an excerpt of two BNL internal reports, in which many details of the methods and the results can be found, as well as detailed references  [2, 3].

3.11.1 FFA Recirculation Loop, Synchrotron Radiation, Polarization

The structure of the FFA loop is as follows:

  • - There are 6 arcs and 6 long straight sections, following RHIC 6-periodicity (Fig. 1).

  • - An arc is comprised of 102 identical QF-drift-BD-drift cells (Fig. 2) wih QF a pure quadrupole and BD a combined function dipole.

  • - Five of the six long straight sections (LSS) are identical and each comprises a string of 52 FODO cells with all energies sharing a common optical axis, coinciding with the quadrupole axes.

  • - Ten dispersion suppression sections (DS) between the arcs and the five LSS are each made up of 18 FFA cells which gradually shift each beam from the LSS axis to its FFA orbit in the arc.

  • - The remaining LSS (RHIC’s IR2 region) is occupied by the 120 m, 42 cavity linac and by the spreader and merger lines at its ends.

  • - Both start and end points of an arc are at the center of a BD magnet in these simulations, for convenience.

  • - The 12 spreader lines at their downstream end, as well as the 12 merger lines at their upstream end, are matched to the 12 sets of FFA orbit optical functions at the center of the arc cell BD magnet.

  • - The spreader at its upstream end and the merger at its downstream end are matched to the optical functions at linac ends.

  • - Path length adjustments (since path length is energy dependent in the FFA arcs) are taken care of in the spreader and merger sections.

  • - In addition, some artefacts are introduced regarding 6D positioning of the bunch at the entrance to the various FFA loop sections. These will be discussed at the appropriate points in the article.

Arc cell

The optical properties of the cell are summarized in four of the figures.

  • - Figures 2 and 3 show respectively the transverse excursion of, and magnetic field along, periodic orbits across the arc cell for the 12 recirculated energies. It can be seen that the field varies in a substantial fashion along the orbit inside a quadrupole at large excursion.

  • - Figure 4 shows the energy dependence of tunes and chromaticities (in this figure \( a\gamma \) is the spin precession rate, with \( a=1.16\times 10^{-3} \) the electron anomalous gyromagnetic factor).

  • - Figure 5 shows the optical functions across the cell at the lower and higher energies.

Synchrotron radiation

Turn-by-turn tracking is performed here, with neither linac nor any spreader and merger sections. The FFA recirculating loop in this first approach is 6-periodic, and perfect fields and perfect optical alignment are assumed.

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Figure 6: Orbits along the FFA recirculation loop for the 12 energies consid- ered: from bottom to top, \( E \)=6.622 to 21.164 GeV in steps of 1.322 GeV. Along the six long straight sections, all energies have a common optical axis (\( x=0 \)). In the arcs the orbit excursion is recorded in the QF\( \rightarrow \)BD drift. The \( \lesssim \)mm excursion observed here at all energies (the largest being at higher energy) is due to a residual orbit mismatch at the dispersion suppressors.

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Figure 7: Energy dependence of the single-turn average energy loss (left axis) and energy spread (right axis) in FFA recirculation loop at the 12 recirculated energies (empty triangles). The solid lines (labelled “theor.”) are from Eq. (1) for a \( 6\times 120 \) cell ring. The slightly lower SR loss from tracking comes from the smaller number of cells per arc, 102, and lower SR loss in the 18-cell DS sections.

Orbit

Figure 6 shows the twelve orbits along the FFA return loop (actually, a plot of the centroid position \( \overline {x(s_f)} \) of a 5000-particle bunch), for \( E=6.622 \) GeV to \( E=21.164 \) GeV in steps of 1.322 GeV. The 12 orbits are all aligned on \( x=0 \) along the six long straight sections, whereas in the six arcs they feature an energy dependent excursion (as seen above in Fig. 2).

An orbit oscillation of \( \lesssim \pm 1 \) mm is visible in the arcs and is also present in the complete ERL simulation (see Fig. 16). This is due to a slight orbit mis-match between LSS and arc across the DS sections. It can be reduced but the main point is what level can be sustained to avoid chromaticity-induced emittance increase  [2].

Synchrotron radiation loss

Figure 7 shows the energy dependence of the synchrotron radiation (SR) induced energy loss and spreading over the 6-period, 3.8 km, FFA loop. These quantities are obtained by a one-turn tracking of a 5000-particle bunch with initial null 6D emittance. The theoretical (“theor.”) average energy loss and energy spread in the figure are obtained using  [2]

(1) \{begin}{align} \label {EqEloss3} \overline {\strut \Delta E}\,[MeV]& = 0.96\times
10^{-15}\,\gamma ^4\left (\dfrac {l_\mathrm {BD}}{\rho _\mathrm {BD}^2}+ \dfrac {l_\mathrm {QF}}{\rho
_\mathrm {QF}^2}\right )\ \times 6~\mathrm {arcs}\times 120~\mathrm {cells/arc}\\ \sigma _E &\approx
1.94 \times 10^{-14}\,\gamma ^{7/2}\sqrt {\dfrac {l_\mathrm {BD}}{|\rho _\mathrm {BD}^3|}+ \frac {l_\mathrm
{QF}}{|\rho _\mathrm {QF}^3|}}\ \times \sqrt {6\times 120},\nonumber \{end}{align}

with 120 the number of cells (as discussed in §3.11.1) necessary to close a circle given the 8.73 mrad single cell deviation (in the FFA loop, the orbit closure is ensured with 102 cells per arc and 18 cells per DS section), and with \( l_\mathrm {QF} \), \( l_\mathrm {BD} \) the magnet lengths and \( \rho _\mathrm {QF} \), \( \rho _\mathrm {BD} \) their average curvature radii as obtained from the stepwise ray-tracing.

Polarization

The spin vector is injected horizontally in the ERL, and precesses around the vertical magnetic field at a rate of \( a\gamma \alpha \) (where \( \alpha \) is the azimuthal angle) in the course of a recirculation round the FFA loop. The 1.322 GeV energy increment ensures a polarization vector parallel to the longitudinal axis at IP6 and IP8. Due to energy spread, spin precession undergoes spreading (“spin diffusion”).

In the following we first assess the effect of SR induced energy spread on spin diffusion, then we assess spin diffusion for a bunch with nominal transverse emittances and momentum spread.

A theoretical approach can be used to check tracking results as follows. The solution of the diffusion equations in constant magnetic field can be written  [4]

\[ \begin {pmatrix}\overline {\strut \Delta E^2}\\[1ex] \overline {\strut \Delta E\Delta \phi
}\\[1ex] \overline {\strut \Delta \phi ^2} \end {pmatrix}= \begin {pmatrix} 1 & 0 & 0\\[1ex] \alpha
s & 1 & 0\\[1ex] \alpha ^2s^2 & 2\alpha s & 1\end {pmatrix} \begin {pmatrix}\overline
{\strut \Delta E^2} \\[1ex] \overline {\strut \Delta E\Delta \phi } \\[1ex] \overline {\strut \Delta \phi
^2} \end {pmatrix}_{s=0} +\omega \times \begin {pmatrix} s \\[1ex] \alpha s^2/2 \\[1ex] \alpha ^2s^3/3 \end
{pmatrix} \]

where \( s \) is the distance in the field, \( \omega = \displaystyle \frac {C}{\rho ^3}{\lambdabar }_\mathrm {c}r_e\gamma ^5E^2 \approx
1.44\times 10^{-27}\,\dfrac {\gamma ^5}{\rho ^3}E^2 \), \( \alpha =\displaystyle \frac {a}{\rho E_0}\approx \dfrac 1{0.4406\rho } \) (with \( \lambdabar _\mathrm {c}=\hbar /m_ec \) the electron Compton wavelength, \( C=110\sqrt {3}/144 \), \( E_0=m_ec^2/e \) the electron rest mass).

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Figure 8: Turn-by-turn in the FFA loop: final polarization \( \langle \cos \Delta \phi \rangle \) (left axis) and spin diffusion \( \sigma _{\phi } \) (right axis) in a 5000-particle bunch (zero size at start of a turn), for the 12 different energies 6.622 to 21.164 GeV in steps of 1.322 GeV.

Assuming a starting state

\[ \begin {pmatrix}\overline {\strut \Delta E^2} \\ \overline {\strut \Delta E\Delta \phi } \\
\overline {\strut \Delta \phi ^2}\end {pmatrix}_{s=0}=0, \]

which is the case for each energy for instance, in the turn-by-turn tracking, yields

\[ \sigma _{E}=\overline {\strut \Delta E^2}^{1/2}=\sqrt {\omega s} \]

(which we note is consistent with the familiar \( \sigma _E/E=3.8\,10^{-14} \dfrac {\gamma ^{5/2}}{\rho ^{3/2}}\sqrt {s} \) ), so that

(3.2--3) \{begin}{gather} \label {EqSphiSE1} \sigma _{\phi }=\overline {\Delta \phi
^2}^{1/2}=\sqrt {\frac {\omega \alpha ^2s^3}3}=\frac {\alpha s} {\sqrt {3}}\,\sigma _{E}\intertext {or,
given $s=2\pi \rho $,}\frac {\sigma _{\phi }} {\sigma _E}=8.23\,\textrm {[rad/GeV/turn]\label {EqSphiSE2}}
\{end}{gather}

Synchrotron radiation effects, turn-by-turn

Spin is tracked turn-by-turn in the 6-period ring as above, namely,

\[ \mathrm {6~\times ~\left [\,DS\ -\ LSS\ -\ DS\ -\ ARC\,\right ]}, \]

with neither linac nor spreader and merger sections.

Tracking results are displayed in Figure 8. The “\( \sigma _E \)” curve is that of Fig. 7, for comparison with the spin diffusion angle rms value, \( \sigma _{\phi } \). Their ratio takes a quasi-constant value \( \sigma _{\phi } /\sigma _E\approx 10 \) rad/GeV close to the expected 8.23 rad/GeV indicated by Equations (2) and (3)). Note that \( a\gamma \alpha \) in this plot appears to differ from an integer multiple of 360 degs (its expected value) by \( \sim \)1-2 degs. This stems from the lack of accuracy of SR energy loss compensation at the linac boost, and is of marginal effect here.

An up-down ER cycle, 23 passes, in a six-arc FFA ring

In order to get a sense of orders of magnitude, we conclude this introduction to the FFA loop with an up-down tracking in a model 6-arc ring, comprising 6\( \times \)120 cells, with, at a single location, a simplified linac simulation using a thin-lens 1.322 GeV boost.

A particle bunch is, in a row, accelerated in 11 linac passes (12 recirculation loops) from 6.622 to 21.164 GeV and decelerated in 11 linac passes (10 recirculation loops) back down to 6.622 GeV. The following features are included in the simulation:

After each turn, prior to tackling the next one,

  • i) SR loss is compensated at the linac by giving a turn-dependent energy kick \( 1.322+\Delta E \) with \( \Delta E \) computed from Eq. (1);

  • ii) the bunch is re-centered in position and angle on the theoretical FFA orbit once per turn, next to the boost (according to the orbit dependence on energy).

Two simulations are performed:

  • i) The first simulation starts with zero 6D emittance (a point object) and produces evolution of the horizontal and longitudinal emittances as displayed in Fig. 9. The vertical emittance remains zero because the photon recoil is not accounted for in the Monte Carlo SR simulation.

    Details of phase space portraits at the various energies are postponed to the complete ERL tracking in §3.11.3, as the present phase space portraits differ only slightly from the full ERL simulation results.

  • ii) The second simulation is carried out with a nominal starting bunch emittance of \( \sim 50\,\pi \mu \)m normalized in both transverse planes, a random, uniform momentum spread over \( \pm 3\times 10^{-4} \), and zero bunch length. The evolution of the horizontal and longitudinal emittances are displayed in Figs. 10 and 11 respectively. In this simulation, different numbers of particles have been tried to test the convergence (1k, 5k and 10k), as well as two different integration step sizes in the two quadrupoles (1 cm and 3 cm). The relative effect is small, the difference is essentially a slight translation of the curves.

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Figure 9: The markers in this figure show the evolution of horizontal (left vertical axis) and longitudinal (right axis) bunch emittances under the effect of SR in the case of an initial point object (zero 6D emittance), over a 23-loop end-to-end up-down cycle in a simplified 6-arc ring (6.622 \( \stackrel {up ~ }{\rightarrow } \) 21.164 \( \stackrel {down}{\rightarrow } \) 6.622 GeV).

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Figure 10: Markers in this figure give the evolution of horizontal (left vertical axis) and vertical (right axis) bunch emittances under the effect of SR for initial conditions \( \epsilon _x=\epsilon _y=50\,\pi \mu \)m normalized, \( \Delta E/E \) uniform random in \( \pm 3\,10^{-4} \) and \( \sigma _l=0 \). The bunch is tracked over a 23-loop end-to-end up-down cycle in a simplified 6-arc ring (6.622 \( \stackrel {up ~ }{\rightarrow } \) 21.164 \( \stackrel {down}{\rightarrow } \) 6.622 GeV). The various curves correspond to either a different number of tracked particles (1, 5 or 10 \( \times 10^3 \)), or to different integration step sizes in cell quadrupoles (1 or 3 cm).

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Figure 11: Evolution of longitudinal bunch emittance (left axis, left four markers) and \( \sigma _l \), \( \sigma _E \) (right axis, right two markers), for initial conditions \( \epsilon _x=\epsilon _y=50\,\pi \mu \)m normalized, \( \Delta E/E \) uniform random in \( \pm 3\times 10^{-4} \) and \( \sigma _l=0 \). The bunch is tracked over a 23-loop end-to-end, up-down cycle in a simplified 6-arc ring (6.622 \( \stackrel {up ~ }{\rightarrow } \) 21.164 \( \stackrel {down}{\rightarrow } \) 6.622 GeV).

We conclude with spin tracking in the previous 6-arc FFA ring, made up of 6\( \times \)120 cells, with, at a single location, a linac simulation by a thin-lens 1.322 GeV boost. A 5000-particle bunch is taken from 6.622 to 21.164 GeV in 11 linac passes. The results are displayed in Figure 12. The cumulated effect amounts to \( \sigma _{\phi }\approx 15 \) degs at the end of the final 21.1 GeV loop (top-left plot). The top-right and bottom plots show that the factor with the dominant influence on the final polarization is the injected bunch energy spread.

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Figure 12: Evolution of cumulated spin diffusion in the case of an 11 linac- pass acceleration cycle (12 complete 6-arc loops), 6.622 \( \rightarrow \) 21.164 GeV, in a simplified 6-arc ring.

3.11.2 ERL optics, complete

This section briefly introduces the handling of the three additional structures needed, namely, the linac, spreader, and merger sections. It concludes with the complete ERL optics, 23 passes.

Linac

Transport through the linac cavities uses “Chambers matrices”; the corresponding source code has been copied from the Saclay code BETA  [5] for reliability.

These matrices take the following form:

For both planes (for either \( x \) or \( y \)):

(4) \begin{equation} \label {EqCham1} \begin {pmatrix} x \\[1ex] x' \end {pmatrix}_\mathrm {out}=
\begin {pmatrix} \cos u-\sqrt {2}\sin u\cos \phi & v W_\mathrm {i}\sin u\cos \phi \\[1ex] -\dfrac {\sin
u}{vW_\mathrm {o}}\big (2\cos \phi +\dfrac 1{\cos \phi }\big ) & \dfrac 1{W_\mathrm {o}W_\mathrm {i}}
\big (\cos u+\sqrt {2}\sin u\cos \phi \big )\end {pmatrix} \begin {pmatrix}x \\[1ex] x' \end
{pmatrix}_\mathrm {in} \end{equation}

with \( u=\log (W_\mathrm {o}/W_\mathrm {i})/(\sqrt {8}\cos \phi ) \), \( v=\sqrt {8}L_\mathrm {cav}/ (W_\mathrm {o}-W_\mathrm {i}) \), \( W_\mathrm {i} \), \( W_\mathrm {o} \) respectively the incoming and outgoing kinetic energies, \( L_\mathrm {cav} \) the cavity length and \( \phi \) the particle phase at the cavity.

If \( (W_\mathrm {o}-W_\mathrm {i})/W_\mathrm {i} \ll 1 \) the matrix is used in the simplified form

(5) \begin{equation} \begin {pmatrix} x \\[1ex] x' \end {pmatrix}_\mathrm {out} = \begin {pmatrix}
\sqrt {W_\mathrm {i}/W_\mathrm {o}} & L_\mathrm {cav}\times \sqrt {W_\mathrm {i}/W_\mathrm {o}} \\[1ex]
0 & \sqrt {W_\mathrm {i}/W_\mathrm {o}} \end {pmatrix} \begin {pmatrix}x \\[1ex] x' \end
{pmatrix}_\mathrm {in} \end{equation}

The code works with determinant 1 matrices, obtained by renormalizing the transport coefficients by the square root of the matrix determinant.

Spreader and merger sections

The 12 spreader lines (linac to FFA arc) and 12 merger lines (FFA arc to linac) in the ERL ensure a series of optical functions: orbit positioning, optical matching between linac and FFA loop i.e. beta functions and horizontal dispersion, which is non-zero on the FFA side), path length (as it is energy dependent in the FFA loop) and \( \mathrm {R}_{56} \) adjustments.

In the simulations, for simplicity we use the same design for all spreader and merger lines, scaled to the different rigidities. One consequence is that, except for the 21.164 GeV spreader and merger lines, SR effects as well as spin dynamics cannot be evaluated (the bending radii, possible presence of a vertical chicane, and some other aspects, have to be optimized separately (e.g. SR has to be minimized) for each spreader/merger line).

ERL optics, readiness

The lattice in the up-down ERL tracking simulations has the following form

(3.6--8) \{begin}{align} \label {EqERL} \mathrm {ERL} & = \underbrace { \mathrm
{merger}\,+\hspace {-5ex} \stackrel {\stackrel {\textbf {\small Observation point\strut }}{\downarrow
}}{\phantom {\_}}\hspace {-5ex}\mathrm {linac} + \mathrm {spreader} }_{\text {RHIC IR2 region}\strut }\quad
+\quad \mathrm {FFA} \\[1ex] \intertext {with} \label {EqERLFF} \mathrm {FFA} & =\mathrm {ARC}- \mathrm
{DS}-\frac 12\mathrm {LSS}+\underbrace {\left [\,\dfrac 12\mathrm {LSS}-\mathrm {DS}-\mathrm {ARC}-\mathrm
{DS}- \dfrac 12 \mathrm {LSS}\right ]}_{{\textbf {\small 4 times}}\strut }+\dfrac 12\mathrm {LSS}-\mathrm
{DS}-\mathrm {ARC}\\[1ex] \intertext {and} \label {EqERLArc} \mathrm {ARC} & = 102\ \times \ \left
[\dfrac 12\mathrm {BD}-\mathrm {drift}-\mathrm {QF}-\mathrm {drift}-\dfrac 12\mathrm {BD}\right ]
\{end}{align}

Note in particular, compared to the “simplified 6-arc” simulations in §3.11.1 and §3.11.1, the absence of DS sections in IR2 region in this complete ERL layout (actually not fully complete, see below, but close enough that it delivers a qualitative overview of the ERL model to be eventually achieved and the outcomes to be expected).

Some more details regarding the optical structure in this simulation of the complete ERL are as follows:

  • - An arc is comprised of 102 identical doublet cells (Eq. (8)) with quadrupole optical axes radially shifted by 13.48 mm with respect to one another to ensure 8.73 mrad bending per cell (in keeping with the optical properties described in §3.11.1).

  • - The five long straight sections (LSS) are made up of a string of 52 such cells with quadrupole axes superimposed instead. These LSS are dispersion free and all energies share a common optical axis (as in Fig. 6), aligned on the quadrupole axes.

  • - The dispersion suppressors (DS) between the arcs and each of the five LSS are comprised of 18 of these cells and have quadrupole axes shifting gradually from zero at their LSS end to 13.48 mm at their arc end. Six of these DS take the 23 beams (12 recirculations up, 11 down) from their respective FFA optical axes in the arcs onto their common axis in the downstream LSS. The other six DS have the reverse functionality.

  • - The remaining straight section is occupied by the 120 m, 42 cavity linac and the spreader and merger lines (along RHIC’s IR2 region, see Fig. 1). There are no energy loss or energy spread compensation cavities in the present simulations.

  • - Both start and end points of an arc are at the center of a BD magnet (Eq. (8)), for convenience.

  • - The spreader at its downstream end and the merger at its upstream end:

    • – steer the beam respectively onto and from the (non-zero) FFA orbits,

    • – are matched to the optical functions at the center of the arc cell BD magnet.

  • - The spreader at its upstream end and the merger at its downstream end are matched to the optical functions and dispersion at linac ends.

  • - The beam transport to the IPs at IR6 and IR8 at top energy (21.164 GeV) is not accounted for; instead the 21.164 GeV recirculation is treated like a regular one, simply taking the bunches back to the deceleration phase for energy recovery.

  • - Path length adjustments (path length is energy dependent in the FFA arcs) are taken care of in the spreader and merger sections.

Perfect optical alignment and perfect fields are assumed everywhere. Moreover, artificial 6D positioning of the bunch is introduced at various locations: this will be addressed in detail in due course. Note also, in the following simulations the entrance point to the linac is the starting point of the optical sequence in Zgoubi, the “Observation point” in Eq. (6).

The optics properties are summarized in Figures 13-16.

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Figure 13: This figure shows the 12 recirculated orbits (obtained by tracking a single particle) from 6.622 to 21.164 GeV, a \( 12 \times 3.887 \) km long path. Each of the 12 “steps” in this plot represents a complete ERL turn (Eq. (6)), 3.887 km long. In the arcs the orbit behaves as detailed in Fig. 6, with excursion ranging from \( \sim \!-1.35 \) cm at 6.622 GeV (left hand end) to \( \sim \!+0.9 \) cm at 21.164 GeV (right hand end). In the five long straight sections between the arcs and in the linac straight between two “steps”, the orbit coincides with the \( x=0 \) axis in the figure.

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Figure 14: This figure shows the betatron functions from 5.3 to 21.164 GeV (computed from the transport of 11 sample particles).The spreader and merger sections correspond to the \( \sim \)200 m spikes. The 120 m long linac section cannot be distinguished, squeezed between spreader and merger lines (see Figs. 1516); it has be- tatron function values \( \beta _x=\beta _y=120 \) m at both ends. The 12 regions between the spikes are along the FFA recirculating loop, where betatron functions increase from \( \beta _x/\beta _y= \)0.51/6.61 m amplitude at 6.622 GeV (leftmost 3.887 km section on the figure) to \( \beta _x/\beta _y=3.57/26.1 \) m at 21.164 GeV (rightmost). The right vertical axis is for the dispersion functions; \( D_y \) is non-zero along short chicane segments only, in the spreader and merger lines, \( D_x \) is in the few centimetres range; the small \( D_x \) oscillation from 15 km on is due to a slight mismatch and is a very small effect (\( \lesssim 10 \) cm, see Fig. 16).

5.3 to 6.622 GeV linac and spreader optics

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Figure 15: Details of the optical functions (betatron, left axis, and dispersion, right axis), in the case of the 5.3 \( \rightarrow \) 6.622 GeV linac energy step. The \( \beta _x \), \( \beta _y \) parabolas on the left correspond to the linac (120 m long). The linac is followed by a spreader line which ends up steering the beam on its 6.622 GeV orbit in the FFA loop on the way up. The FFA loop extends into the \( s>230 \) m region (to the right), with betatron amplitudes \( \beta _x,\ \beta _y=5.6,\ 6.6 \) m and dispersion function amplitude -5 to +3 mm (see Fig. 5). The latter features \( \sim \!\pm 1.5 \) m excursion in the spreader.

9.842 to 21.164 GeV merger-linac-spreader optics

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Figure 16: Details of the optical functions (betatron, left axis, and disper- sion, right axis), in the 19.842 \( \rightarrow \) 21.164 GeV region. The \( \beta _x \), \( \beta _y \) parabolas in the middle region are in the linac. The linac section is preceded by a merger and followed by a spreader line, with, upstream and downstream of the latter, beam steering from and onto its respective 19.842 and 21.164 GeV orbits in the FFA loop. The FFA loop extends into the \( s<42600 \) m and \( s>42950 \) m regions, with vertical betatron amplitudes \( \beta _y=20 \) m to the left, \( \beta _y=26 \) m to the right and, superimposed, an oscillation resulting from cumulated mismatch. The dispersion function has \( \pm 10 \) cm oscillation in the FFA loop, due to cumulated upstream mismatch.

3.11.3 Tracking the ERL

In this concluding section, the full ERL layout is considered, with optical settings as discussed in §3.11.2:

\[ \mathrm {ERL} = \underbrace { \mathrm {merger}\,+\hspace {-5ex} \stackrel {\stackrel {\textbf
{\small Observation point\strut }}{\downarrow }}{\phantom {\_}}\hspace {-5ex}\mathrm {linac}+\mathrm
{spreader} }_{\text {RHIC IR2 region}\strut }\quad +\quad \mathrm {FFA} \]

with FFA as in Eq. (7). As pointed out earlier, some artefacts and limitations are imposed on the modeling of the ERL at this stage of its development, as follows.

Artefacts
  • \( \bullet \) Artificial bunch centroid centring is applied along the ERL (using Zgoubi’s “AUTOREF” keyword  [7]), as follows:

    • - on exit of any of the 12 merger lines (i.e., at the entrance to the linac):

      • (i) horizontal \( (x,x') \) and vertical \( (y,y') \) bunch centring on zero (a substitute to beam steering onto the linac optical axis);

      • (ii) bunch centring on the design momentum (this stands for artificial compensation of the SR loss that occurs in the upstream FFA arc and merger line);

      • (iii) time centring so that at any stage in the acceleration-deceleration cycle bunches will enter the linac centered on the RF crest.

    • - on exit of any of the 12 spreader lines there is bunch centring on the current FFA orbit (a substitute for beam steering), centring on the design momentum (this stands for artificial compensation of SR loss in the spreader);

    • - at the entrance to each of the five LSS (i.e., going from arc to straight) there is horizontal \( (x,x') \) and vertical \( (y,y') \) bunch centring on zero (this cancels SR induced orbit in the arcs and induced orbit by the DS section).

  • \( \bullet \) Limitations in the model in relation with these artefacts and with other approximations which they entail, include:

    • - SR is switched off in all spreader and merger lines (and only there) except in the top energy spreader and merger lines at 21.164 GeV. As a consequence, except for the latter, their contributions to SR induced energy losses and related beam and spin dynamics effects are not accounted for;

    • - The same conditions for spin tracking: switched off in all 6.622 to 19.842 GeV spreader and merger lines (and only there).

Way up, 6.622 to 21.164 GeV
Beam ellipses at the linac ends

Correct behaviour of the tracking is first assessed at the linac ends: one hundred particles evenly distributed on the paraxial invariant with \( \beta =120 \) m, \( \alpha =\pm 1 \) (both horizontal and vertical) are launched at the linac entrance with \( E \)=5.3 GeV, for a 12 linac-pass tracking up to 21.164 GeV. Betatron damping has been inhibited in this case (Chambers matrices as in Eq. (4) are normalized to have unit determinant).

Tracking shows that beam ellipse parameters remain at \( \beta =120 \) m, \( \alpha =\pm 1 \) at a few % level at both linac ends, in both planes, all the way from 5.3 to 21.164 GeV, see Fig. 17.

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Figure 17: The figure shows a superimposition of 12, 100-particle bunches, at the linac entrance (each 100 particle set is spread on a converging ellipse, all 12 ellipses do superimpose) and the same bunches at the linac exit (each 100-particle set is spread on a diverging ellipse, all 12 ellipses do superimpose).

\( (x,x') \) AT LINAC ENDS

5.300 and 6.622 GeV

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10.5 and 11.9 GeV

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19.8 and 21.164 GeV

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\( (y,y') \) AT LINAC ENDS

5.300 and 6.622 GeV

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10.5 and 11.9 GeV

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19.8 and 21.164 GeV

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Figure 18: Bunch transverse emittances (non-normalized) and \( \beta _x,\alpha _x \), \( \beta _y,\alpha _y \) parameters are given below the plots for each energy. These parameters appear to be well preserved, with \( \beta \gamma \) betatron damping as expected, i.e. the same normalized emittances at the linac entrance and exit.

5000-particle bunches at linac ends

A 5000-particle bunch is tracked here. We show that transverse and longitudinal bunch emittances, as observed at the linac ends, behave in a reasonable manner. The details still require further investigation.

Initial bunch emittances at 5.3 GeV are 23\( \pi \mu \) m transverse, zero longitudinal (both length and \( \Delta E/E \) zero). Linac damping is ataken into account as is synchrotron radiation. The results are displayed in Figures 18 and 19.

Longitudinal Phase Space at LINAC EXIT

(initial bunch length and energy spread zero)

7.9 GeV

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9.2 GeV

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10.5 GeV

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11.9 GeV

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21.164 GeV

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21.164 GeV centered

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Figure 19: The bunch longitudinal rms emittance (in \( \mu \)eV.s) is given at the bottom of each plot for each energy. It appears to behave reasonably well (simulation- wise) over the full energy range from 5.3 to 21.16 GeV. Initially zero (at 5.3 GeV), \( \epsilon _l \) remains small at top energy (bottom right plot); growth mechanisms include synchrotron radiation. There remains a need for further investigation.

Up-down cycle in the FFA stage ERL
Linac damping off and SR off

In order to ensure that input data files for the 23 linac passes end-to-end tracking are set up correctly, a preliminary up-down cycle is performed with linac damping off and synchrotron radiation off. A 2000 particle bunch is tracked with initial bunch emittances and longitudinal parameters

\[ \epsilon _{x,\,\mathrm {norm}}=\epsilon _{y,\,\mathrm {norm}}=23\,\pi \,\mu \mathrm {m},\qquad
\sigma _l=0, \qquad \sigma _E=0\,. \]

Transverse emittances are expected to be preserved, and longitudinal beam size growth is expected to be commensurate with SR-induced growth observed in the case of the 6-arc model, §3.11.1.

Tracking results are displayed in Figs. 20 and 21. In Fig. 20, a particle is represented by an empty box marker. It can be seen that at each energy the 2000 boxes superimpose perfectly - at that scale. Fig. 21 shows phase space details at the end of the acceleration-deceleration cycle, when the energy has returned to 5.3 GeV. This tracking demonstrates the preservation of the orbits and of the transverse emittances, and small longitudinal emittance growth, over a complete \( 5.3\,\stackrel {up}{\rightarrow }\,21.164\,\stackrel {down}{\rightarrow }\,5.3 \) GeV cycle.

Note that no symplecticity issue is expected: the tracking distance here is very short compared to routinely hundreds of thousands of turns tracked for proton polarization studies in RHIC, using a similar integration step size and non-linear optics.

Linac damping and SR on

An important aspect at this stage is that there has been no optimization regarding bunch transmission. This is beyond the scope of the present work which concerns the setting up of the data and data files for end-to-end simulation studies.

That said, tracking is performed here with synchrotron radiation and with unnormalized Chambers matrices, i.e., betatron damping is accounted for. The results are displayed and commented on in Figs. 22 and 23.

Fig. 22 shows that the bunch undergoes noticeable energy spreading (on the scale of the figure) beyond pass \( 18\sim \,19 \) (when the markers no longer superimpose).

Transverse emittance growth observed in Fig. 23 and requires further investigation. A large sine-like distortion of the bunch in longitudinal phase space at the final energy after deceleration (5.3 GeV) can be seen in the bottom plot in Fig. 23. This can be compared with the SR-free case, the bottom plot in Fig. 21.

Fig. 24 shows the evolution of SR energy loss over 23 recirculations from 5.3 to 21.1 GeV and back to 5.3 GeV.

Note that, as mentioned in §3.11.3, bunches always present themselves at the RF crest at the linac entrance.

SR off, DAMPING off

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Figure 20: Average kinetic energy of a 2000-particle bunch at entrance and exit of the linac (hence two markers per pass, aligned vertically), as a function of pass number (each particle is represented by an empty box). What appears as one box is actually a superimposition of 2000 boxes, so the bunch is well confined from 5.3 GeV at injection to 21.164 GeV and back down to 5.3 GeV.

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Figure 21: Phase spaces back to 5.3 GeV, horizontal (top), vertical (middle) and longitudinal (bottom). The former two feature a preserved 23 \( \mu \)m normalized emittance, the latter shows a very small final longitudinal emittance.

       

SR on, DAMPING on

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Figure 22: Average kinetic energy of the 2000-particle bunch, at entrance and exit of the linac (hence two markers per pass), as a function of pass number (each particle is represented by an empty box). The bunch appears to undergo serious energy spreading from pass \( 12\sim 13 \) on to final 5.3 GeV.

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Figure 23: Phase spaces at 5.3 GeV after 23 recirculations, hori- zontal (top), vertical (middle) and longitudinal (bottom). The effects of SR are substantial.

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Figure 24: Evolution of SR energy loss over 23 recirculations from 5.3 to 21.1 GeV and back to 5.3 GeV. There are various reasons for the non-symmetry of the “per pass” curve with respect to pass number 12; the dominant cause needs to be investigated.

Polarization

The polarization state out of these simulation, for a 5000 particle bunch at top energy after acceleration from 5.3 to 21.164 GeV, is displayed in Fig. 25, in both SR off and SR on cases.

Bunch polarization at collision energy, 21.164 GeV

SR off:

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SR on:

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Figure 25: Bunch polarization states (for 5000 particles) at top energy, SR off (top row) and SR on (bottom row).

References