Computational Considerations for Spin Tracking in Electron-Ion Colliders

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1 Computational Considerations for Spin Tracking in Electron-Ion Colliders Dan T Abell RadiaSoft, LLC, Boulder, CO dabell@radiasoftnet JLEIC Collaboration Meeting 3 5 April 27

2 The Goal of Spin Tracking is Insight lumi-opt / 3 / 85 GeV / Q y = / SODOM P lim$ $ 2 [Q y ] 8 $ P lim & y / # mm mrad M Vogt, Bounds on the Maximum Attainable Equilibrium Polarization of Protons at High Energy in HERA, PhD dissertation, Universität Hamburg, 2 2

3 The Goal of Spin Tracking is Insight a: HERA-p / 8 snakes / 4 pi mm mrad / 8 GeV b: HERA-p / 8 snakes / 4 pi mm mrad / 82 GeV Figure : The ˆn vector for the 4π mm mrad ellipse at 8 GeV (left) and 82 GeV (right) a: HERA-p / 8 snakes / 64 pi mm mrad / 8 GeV b: HERA-p / 8 snakes / 64 pi mm mrad / 82 GeV DP Barber, K Heinemann, GH Hoffstaetter, and M Vogt, arxiv:physics/9942, Jan 999 Figure 2: The ˆn vector for the 64π mm mrad ellipse at 8 GeV (left) and 82 GeV (right) 3

4 The Accuracy of Orbital Tracking Affects Spin Tracking Xn b m 2 4 DKD n b 8 Orbit DKD BKMK n b Xtm 6 8 DKD n b 4, retuned BKMK n b BKMK n b t Spin Sn b S ref n b DKD BKMK no Romberg StS ref DKD n b 8 DKD n b 4, retuned BKMK no Romberg BKMK n b t 4

5 How You Integrate the Quadrupole Affects the Tune H Q = s + 2 b P t + P 2 t P 2 x P 2 y + b P t drift = 2 (P2 x + P 2 y )+ k 2 2 (X2 Y 2 ) matrix = 2 s Px 2 + Py 2 q + b 2 P t + Pt 2 matrix + k 2 2 (X2 Y 2 ) + k 2 2 (X2 Y 2 ) kick Drift-Kick + 2 b P t + P 2 t P 2 x P 2 y + b P t 2 (P2 x + P 2 y ) kick s + 2 b P t + P 2 t P 2 x P 2 y + b P t 2 Matrix-Kick Matrix -Kick Px 2 + Py 2 q + b 2 P t + Pt 2 kick with P t = mgc2 mg c 2 p c = g g b g Symplectic integration made easy: (i) Split the Hamiltonian into solvable parts (ii) Construct a symmetric mapping of the parts Voilà: a second-order symplectic map! 5

6 How You Integrate the Quadrupole Affects the Tune H Q = s + 2 b P t + P 2 t P 2 x P 2 y + b P t drift = 2 (P2 x + P 2 y )+ k 2 2 (X2 Y 2 ) matrix = 2 s Px 2 + Py 2 q + b 2 P t + Pt 2 matrix + k 2 2 (X2 Y 2 ) + k 2 2 (X2 Y 2 ) kick Drift-Kick + 2 b P t + P 2 t P 2 x P 2 y + b P t 2 (P2 x + P 2 y ) kick s + 2 b P t + P 2 t P 2 x P 2 y + b P t 2 Matrix-Kick Matrix -Kick Px 2 + Py 2 q + b 2 P t + Pt 2 kick with P t = mgc2 mg c 2 p c = g g b g This splitting does not require fitting the quad strengths Moreover, it yields correct tunes for off-energy particles 6

7 Use Quaternions for Integrating the Spin Motion element ~W ~ W ~ W2 ~ W3 ~ W4 orbital slice quaternion for crossing this element h Q(h) =q( 2 h ~ W4 ) q(h ~ W3 ) q(h ~ W2 ) q(h ~ W ) q( 2 h ~ W ) 2nd-order convergence for the spin motion Then use Romberg quadratures to accelerate the spin convergence S N S best k k 2 k 3 PWC quadrupole S N S best k k 2 PWC sbend N slices 6 4 N slices 7

8 Use Quaternions for Integrating the Spin Motion (cont) ~W ~ W2 ~ W4 2h Q(h) =q( 2 h ~ W4 ) q(h ~ W3 ) q(h ~ W2 ) q(h ~ W ) q( 2 h ~ W ) Q(2h) =q(h ~ W4 ) q(2h ~ W2 ) q(h ~ W ) 8

9 Use Quaternions for Integrating the Spin Motion (cont) ~W ~ W4 4h Q(h) =q( 2 h ~ W4 ) q(h ~ W3 ) q(h ~ W2 ) q(h ~ W ) q( 2 h ~ W ) Q(2h) =q(h ~ W4 ) q(2h ~ W2 ) q(h ~ W ) Q(4h) =q(2h ~ W4 ) q(2h ~ W ) 9

10 Use Romberg Quadratures to Accelerate the Convergence ~W ~ W ~ W2 ~ W3 ~ W4 h R 2 = Q(h) =q( 2 h ~ W4 ) q(h ~ W3 ) q(h ~ W2 ) q(h ~ W ) q( 2 h ~ W ) R = Q(2h) =q(h ~ W4 ) q(2h ~ W2 ) q(h ~ W ) R = Q(4h) =q(2h ~ W4 ) q(2h ~ W ) Romberg Tableau R R R = 4R R 4 R 2 R 2 = 4R 2 R 4 R 22 = 42 R 2 R 4 2 general term R j+,k = 4j+ R jk R j,k 4 j+

11 Invariant Spin Field Restricts Equilibrium Polarization S z one turn map S2 = R(z ) S z 2 = Mz

12 Use Stroboscopic (Ergodic) Averaging to Compute the ISF average all the spins tracked back to z R q k ŷ z, q z 2, q 2 z k, q k z 4, q 4 z 3, q 3 Stroboscopic averaging yields a non-perturbative computation K Heinemann and GH Hoffstätter, Tracking algorithm for the stable spin polarization field in storage rings using stroboscopic averaging Phys Rev E 54(4): , Oct 996 z, q but 2

13 Convergence can be Slow at Large Angles 2 5 n = 2725, q i err = 85 θerr / rad N turns It should be possible to speed this up dramatically by averaging over a small number of turns and then using that result as a starting point for further computation See Dave Sagan's BMad 3

14 Summary The goal of spin tracking is insight The accuracy of orbital tracking affects the accuracy of spin tracking Avoid any scheme that requires you to refit the quad strengths Symplectic tracking is important even for electrons The invariant spin field tells us best we can achieve Look at the ISF first to understand where you want to end up Then investigate acceleration 4

15 Thank you! Many thanks also to DP Barber, D Meiser, VH Ranjbar, DC Sagan, and F Méot Supported in part by the US Department of Energy, Office of Science, Office of Nuclear Physics, including Award No DE-SC78

Higher-order effects in polarized proton dynamics

PHYSICAL REVIEW SPECIAL TOPICS - ACCELERATORS AND BEAMS, VOLUME 2, 4 (999) Higher-order effects in polarized proton dynamics G. H. Hoffstaetter* Institute for Applied Physics, Darmstadt University of Technology,