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

2. 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

3. 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

4. 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

5. 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

6. 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 = cm
Q2 = cm
Q3 = cm
1st-order matrix:
xf xf’ yf yf’ tf xi
xi’ yi
yi’ ti qi
X-motion
Y-motion

7. Final Focusing Block 2 conditions: M(xf |xi) = 0, M(yf |yi) = 0
2 parameters: Q4  cm, Q5  cm
To have large y at the final quad’s exit: Q6  cm
1st-order matrix:
xf xf’ yf yf’ tf xi
xi’ yi
yi’ ti qi

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

9. IR Up to IP

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

11. Sextupole Compensation
Make M(xf | xi qi) = 0 and M(yf | yi qi) = 0 by adjusting s1  cm and s2  cm

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

13. Sextupole Compensation
Assume:
3rd-order aberrations:

14. Octupole Compensation
Minimize M(xf | xi3) and M(yf | yi3) by introducing 2 pairs of octupoles with o1  cm and o2  cm

15. Octupole Compensation
Assume:
3rd-order aberrations:

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

17. 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

18. Dynamic Aperture & Tracking
Assume:

19. Dynamic Aperture & Tracking
Assume:

20. 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

21. 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)