Chromatic Corrections Vasiliy Morozov and Yaroslav Derbenev for the JLab EIC Study Group

Outline Symmetry-based IR design concept IR linear optics design Analysis and compensation of 2nd- and 3rd-order aberration terms contributing to beam smear at IP Chromaticity compensation Dynamic aperture tracking

IR Design Challenges Low * essential to ELIC’s high-luminosity concept Large size of extended beam f * = F2 Large chromatic spread at IP F ~ Fp/p >> * requires sextupole compensation Non-linear field effects must be accounted for

Symmetry Concept Dedicated Chromaticity Compensation Block (CCB) symmetric around IP Model assuming large parallel beam and ignoring angular divergence Take advantage of symmetry to reduce number of compensation conditions Conditions for compensation of 2nd-order aberrations at IP Chromatic spread (x, y are betatron trajectory components) Smear due to betatron beam size and 2nd-order dispersion effects  Satisfied automatically for symmetric x2 and y2 symmetries of D and ns opposite to symmetry of x Compensation of chromatic tune spread chromatic and sextupole beam smear at IP

IR Design Modular approach: IR designed independently to be later integrated into the ring Utilize COSY Infinity calculates coefficients M(x|) of expansion of type to arbitrary order (+++++) for each of coordinate components Design system such that

Chromaticity Compensation Block  3 symmetry requirements: x(sf) = -x(si)  M(xf |xi) = -1 y(sf) = y(si)  M(yf |yi) = 1 D(sf) = D(si)  M(xf |q) = 0 3 parameters: Q1, Q2, Q3 Q1 = -0.174 T @ 5 cm Q2 = 0.350 T @ 5 cm Q3 = -0.079 T @ 5 cm 1st-order matrix: xf xf’ yf yf’ tf xi xi’ yi yi’ ti qi X-motion Y-motion

Final Focusing Block 2 conditions: M(xf |xi) = 0, M(yf |yi) = 0 2 parameters: Q4  -0.308 T @ 5 cm, Q5  0.850 T @ 5 cm To have large y at the final quad’s exit: Q6  -0.750 T @ 5 cm 1st-order matrix: xf xf’ yf yf’ tf xi xi’ yi yi’ ti qi

IR Up to IP xf xf’ yf yf’ tf 1st-order matrix: xi xi’ yi yi’ ti qi

IR Up to IP

Beam Size & 2nd-Order Aberrations Assume: 1st-order matrix Geometric beam size at IP due to emittance 2nd-order aberrations: 3rd-order aberrations negligible

Sextupole Compensation Make M(xf | xi qi) = 0 and M(yf | yi qi) = 0 by adjusting s1  0.055 T @ 5 cm and s2  -0.180 T @ 5 cm

Sextupole Compensation Assume: 1st-order matrix unchanged Geometric beam size at IP due to emittance 2nd-order aberrations:

Sextupole Compensation Assume: 3rd-order aberrations:

Octupole Compensation Minimize M(xf | xi3) and M(yf | yi3) by introducing 2 pairs of octupoles with o1  0.092 T @ 5 cm and o2  -0.146 T @ 5 cm

Octupole Compensation Assume: 3rd-order aberrations:

Chromatic Tune Dependence Up to 5th order in p/p

Dynamic Aperture & Tracking To estimate Dynamic Aperture limitations due to IR note symmetry of IR note that betatron phase advance in each plane is n rather than completing the ring, represent the rest of the ring by linear matrix Track multiple turns through Isolate IR effects

Dynamic Aperture & Tracking Assume:

Dynamic Aperture & Tracking Assume:

Summary Introduced dedicated Chromaticity Compensation Block (CCB) symmetric around IP Arranged CCB’s magnetic structure and orbital motion to meet certain symmetry requirements Demonstrated compensation of leading-term aberrations at IP, largest of a few remaining aberrations is under 10% of beam size Demonstrated chromaticity compensation Dynamic aperture tracking underway with promising results

Next Steps Further optimization Compensation of lowest-order effect of angular divergence on dynamic aperture Larger-scale octupole symmetry (across IP or 2 IP’s) to improve dynamic aperture Integration/matching of IR to the ring Chromaticity compensation and tracking using complete ring Benchmark numeric results against independent code Design ion ring IR (similar but no emittance and polarization degradation issues)