1. ChromaTiCity Compensation in the Ion and Electron Collider Rings
Yuri Nosochkov, Min-Huey Wang (SLAC)
Fanglei Lin, Vasiliy Morozov, Guohui Wei (JLab)
JLEIC Collaboration Meeting Spring 2016 Thomas Jefferson National Accelerator Facility Newport News, VA

2. JLEIC Collaboration Meeting Spring 2016
Introduction
A natural characteristic of a low-b* Interaction Region is a very high b-function in the final focus quadrupoles near the IP. The latter create large chromatic perturbations resulting in a large non-linear momentum variation of betatron tune and beta functions (beam size). These effects may significantly reduce the beam momentum range affecting the beam lifetime and luminosity.
Solution  a local non-linear chromaticity compensation using dedicated sextupoles placed in the nearest to FF dispersive regions.
Such correction must also include compensation of the non-linear geometric (amplitude dependent) aberrations created by these sextupoles.
Several configurations have been investigated starting from the compact scheme (CCB by V. Morozov, Ya. Derbenev, et. al), as well as schemes based on non-interleaved and interleaved –I sextupole pairs.
Below we will recall the features of such compensation, the previous study for the ion ring, and present the recent results for the electron ring.

3. JLEIC Collaboration Meeting Spring 2016
FF chromaticity
FF quads with large beta functions (bq) make a strong chromatic kick resulting in perturbation of the b-function which to 1st order in d=Dp/p propagates as:
This uncorrected perturbation will result in:
beam smear at IP due to momentum dependence of b*(d) and beam size s(d)
non-linear momentum dependence of betatron tune resulting in a larger tune footprint  limiting dynamic aperture due to exposure to more resonances.
Compensation of the FF large sources of non-linear chromaticity requires a dedicated local correction system.
It has been shown analytically that the 1st order Db(d) gives rise to the 2nd order (~d2) chromatic tune shift  Correction of the FF Db(d) will compensate both the IP beam smear and the 2nd order term of chromatic tune shift.
d = 0.15% FF
b(d) after FF

4. Local compensation using dedicated sextupoles
JLEIC Collaboration Meeting Spring 2016
Local compensation using dedicated sextupoles
Requires two sets of dedicated sextupoles (Chromaticity Correction Block) on each side of IR for local and independent correction of each FF.
Cancel the FF Db(d) with an opposite wave created by the sextupoles placed np in phase advance (in the correction plane) from the FF (np+p/2 from IP).
Make large bshs at the sextupoles for a reasonable sextupole strength K2l.
Make large bx/by (or by/bx) ratio at X (Y) sextupoles for orthogonal X & Y correction.
Design the scheme where non-linear amplitude dependent effects from the sextupoles are compensated (example  -I sextupole pairs).
Momentum variation of phase advance m(d) between the sextupoles and FF can make the correction less perfect. Therefore: 1) place the sextupoles close to the FF to minimize chromatic effects between them, and 2) fine tune the sextupole phase advance around np for minimal high order chromatic terms.
sext FF IP
Db/b (d)
∆𝛽 𝛽 ~ − 𝛽 𝑠 𝜂 𝑠 𝐾 2 𝑙𝛿
∆𝛽 𝛽 ~ 𝛽 𝑞 𝐾 1 𝑙𝛿
m = np

5. Sextupole schemes ……. ……. ……. linear chrom sext in arc FF sext -I
JLEIC Collaboration Meeting Spring 2016
Sextupole schemes
linear chrom
sext in arc
FF sext -I
np(x)
mp(y) IP FF
…….
Original CCB with two X and one Y sextupoles. Most compact scheme. High b at the sextupoles. Some residual non-linear geometric effects due to X and Y sextupoles overlap. -I
np(x)
mp(y) IP FF
…….
Two non-interleaved –I sextupole pairs (X & Y). High b at the sextupoles. Cancellation of sextupole geometric effects. -I
np(x)
mp(y) IP FF
…….
Several distributed interleaved –I sextupole pairs. Moderately high b at the sextupoles. Residual geometric sextupole effects due to pairs overlap.
Linear chromaticity correction: Two family interleaved sextupoles (X & Y) in periodic arc cells. Choosing the number of cells with these sextupoles such that their total phase advance is 2pk will compensate the sextupoles 2nd order geometric effects.

6. Ion ring lattice -- without FF sextupoles
JLEIC Collaboration Meeting Spring 2016
Ion ring lattice -- without FF sextupoles
2.2 km 8-figure ring with one IP (baseline)
90° arc
cells
Straight IP
add FF sext

7. IR quadrupole chromaticity in ion ring
JLEIC Collaboration Meeting Spring 2016
IR quadrupole chromaticity in ion ring
Quad linear chromaticity FF xx xy IP
Asymmetric IR optics with L* = 3.6 m / 7.0 m.
bx* = 10 cm, by* = 2 cm, bmax = 2340 m (60 times higher than in arcs).
FF contributes 40-50% to ring linear chromaticity.
No suitable IR dispersion → sextupoles must be placed outside of the IR.

8. CCB – original compact scheme for ion ring
JLEIC Collaboration Meeting Spring 2016
CCB – original compact scheme for ion ring p
mx = mp
my = np SY SX IP
CCB FF
Compact optics, large b and h at sextupoles → weak sextupoles.
However, there are residual non-linear geometric effects (dnx/dJy), caused by interleaved SX, SY sextupoles, that limit dynamic aperture.
Remaining ring linear chromaticity is cancelled with 2-family sextupoles in the other arc cells.

9. Local non-interleaved –I sextupole pairs in ion ring
JLEIC Collaboration Meeting Spring 2016
Local non-interleaved –I sextupole pairs in ion ring
Two non-interleaved –I sextupole pairs on each side of IP; large b-functions and bx(y)/by(x) ratios at the sextupoles.
Non-interleaved –I pairs provide compensation of the sextupole non-linear geometric effects.
Ring tune is optimized in tracking simulations.
1 (x) + 1 (y)
–I pairs
12 (x) + 12 (y)
sextupoles
Y-pair
X-pair
The remaining ring linear chromaticity is canceled using 48 two-family sextupoles in the other arc cells.
The number of cells with these sextupoles is chosen to make multiple of 2p phase advance providing compensation of the 2nd order sextupole geometric effects.
Qx,y = / 23.16 IP

10. Distributed interleaved –I sextupole pairs in ion ring
JLEIC Collaboration Meeting Spring 2016
Distributed interleaved –I sextupole pairs in ion ring
Six interleaved –I sextupole pairs on each side of IP, where b-functions and bx(y)/by(x) ratios at the sextupoles are made moderately large.
Some residual sextupole non-linear geometric effects due to overlap of the –I pairs .
3 (x) + 3 (y)
–I pairs
12 (x) + 12 (y)
sextupoles
3+3 pairs
Qx,y = / 24.16
The remaining ring linear chromaticity is canceled using 48 two-family sextupoles in the other periodic arc cells. IP

11. Compensation of Db(d) in ion ring
JLEIC Collaboration Meeting Spring 2016
Compensation of Db(d) in ion ring
Db(d)/b is defined in MAD through a chromatic W-function.
The strengths of “non-linear” sextupoles are adjusted to cancel W-function at the IP in order to minimize the IP chromatic beam smear.
By optimizing phase at the sextupoles this correction can be made local, so W-function is minimized both at IP and outside of IR.
In practice, a compromise needs to be reached between correction of W-function and non-linear chromatic tune shift.
Compact CCB with optimal sextupole phase IP
non-linear
sextupoles

12. Correction of non-linear chromatic tune shift in ion ring
JLEIC Collaboration Meeting Spring 2016
Correction of non-linear chromatic tune shift in ion ring
Qx,y = / 23.16
Qx,y = / 24.16
Non-interleaved –I pairs
Interleaved distributed –I pairs
Shown for d = ±0.3% ≈ ±10sd
Similar quality compensation (not shown) with the compact CCB scheme.
Linear chromaticity is corrected to +1.
Phase advance between the “non-linear” sextupoles and IP is set to np+p/2.
Compensation is adequate in all 3 schemes.
Further reduction of non-linear dependence may be possible with fine tuning of the sextupole phase advance  reduction of 3rd and higher order terms.

13. Correction of b* chromatic variation in ion ring
JLEIC Collaboration Meeting Spring 2016
Correction of b* chromatic variation in ion ring
Qx,y = / 23.16
Qx,y = / 24.16
Non-interleaved –I pairs
Interleaved distributed –I pairs
Good correction in all three schemes.

14. Impact on dynamic aperture in ion ring
JLEIC Collaboration Meeting Spring 2016
Impact on dynamic aperture in ion ring
Non-interleaved –I pairs
Interleaved distributed –I pairs
at IP
Scheme
Non-interleaved –I pairs
Distributed interleaved –I pairs
Compact CCB
dnx/dJx
-2.87E+02
4.33E+02
1.15E+01
dny/dJy
-5.12E+02
1.74E+03
1.01E+02
dny/dJx
-2.56E+02
-5.70E+03
-1.26E+04
Non-interleaved –I sextupole scheme provides a larger dynamic aperture without magnet errors due to better compensation of sextupole geometric non-linearities.
Magnet non-linear field errors (studied so far) reduce the aperture and make it comparable for the compact CCB and non-interleaved –I pairs schemes.

15. Electron ring lattice – without FF sextupoles
JLEIC Collaboration Meeting Spring 2016
Electron ring lattice – without FF sextupoles
108° arc
cells IP
add FF sext
beam
uncoupled emittance
ex = 8.9 nm
at 5 GeV
x = -113/-120
Fractional tune (.22/.16) as in ion ring

16. IR quadrupole chromaticity in electron ring
JLEIC Collaboration Meeting Spring 2016
IR quadrupole chromaticity in electron ring
Quad linear chromaticity FF xx xy IP
beam
bx* = 10 cm, by* = 2 cm, bmax = 769 m (30 times higher than in arcs).
FF contributes 25-30% to ring linear chromaticity.
No suitable IR dispersion → sextupoles must be placed outside of the IR.

17. Compact CCB scheme for electron ring
JLEIC Collaboration Meeting Spring 2016
Compact CCB scheme for electron ring p SY SX
Compact optics, large b and h at sextupoles.
Some non-linear geometric effects (dnx/dJy), caused by interleaved SX, SY sextupoles, are not compensated which may limit dynamic aperture.
Remaining ring linear chromaticity is cancelled with 120 two family sextupoles in the arc cells.
Emittance is increased ~80% due to increased b and h in the CCB.
Sextupole phase advance and ring tune are set using thin lens phase trombones.

18. Local non-interleaved –I sextupole pairs in electron ring
JLEIC Collaboration Meeting Spring 2016
Local non-interleaved –I sextupole pairs in electron ring
Two non-interleaved –I sextupole pairs on each side of IP (2 arc cells per pair); large b-functions and bx(y)/by(x) ratios at the sextupoles.
Sextupole non-linear geometric effects are compensated.
Remaining ring linear chromaticity is canceled using 80 two-family sextupoles in the 108° arc cells. Number of cells with these sextupoles corresponds to multiple of 2p phase advance providing compensation of the 2nd order sextupole geometric effects.
Uncoupled emittance: ex = 19.5 nm at 5 GeV.
X-pair
Y-pair
Qx,y = / 45.16, x = -127/-145 IP
Sextupole phase and ring tune are set using thin phase trombones

19. Distributed interleaved –I sextupole pairs in electron ring
JLEIC Collaboration Meeting Spring 2016
Distributed interleaved –I sextupole pairs in electron ring
4 interleaved –I sextupole pairs on each side of IP in regular periodic arc cells (five 108° cells per pair)  almost half-arc is needed per each FF.
Residual sextupole non-linear geometric effects due to interleaved sextupoles.
Remaining ring linear chromaticity correction using 80 two-family sextupoles in the other periodic arc cells.
Sextupole phase and ring tune are set using thin phase trombones.
uncoupled emittance
ex = 8.9 nm
at 5 GeV
4 –I pairs
4 –I pairs IP
Qx,y = / 47.16, x = -113/-120

20. Non-interleaved –I pairs for lower emittance in electron ring
JLEIC Collaboration Meeting Spring 2016
Non-interleaved –I pairs for lower emittance in electron ring
The non-interleaved scheme is modified to reduce b-functions at the –I sextupoles by 40% for a lower emittance at the expense of higher sextupole strengths (still reasonable).
Emittance is reduced by 20% to ex = 15.5 nm at 5 GeV.
The other features are identical to the –I scheme with higher b.
X-pair
Y-pair
Qx,y = / 45.16, x = -120/-133 IP

21. Correction of non-linear chromatic tune shift in electron ring
JLEIC Collaboration Meeting Spring 2016
Correction of non-linear chromatic tune shift in electron ring
Only linear chrom sext
Distributed interleaved –I pairs
Compact CCB
Non-interleaved –I pairs
and lower emittance
Shown for d = ±0.4% ≈ ±9sd
Phase advance from the “non-linear” sextupoles to IP is set to np+p/2.
Larger momentum range is achieved with compact CCB and non-interleaved –I pairs.

22. Correction of b* chromatic variation in electron ring
JLEIC Collaboration Meeting Spring 2016
Correction of b* chromatic variation in electron ring
Only linear chrom sext
Distributed interleaved –I pairs
Compact CCB
Non-interleaved –I pairs
and lower emittance
Better correction with compact CCB and non-interleaved –I pairs.

23. JLEIC Collaboration Meeting Spring 2016
Phase optimization at non-interleaved –I sextupole pairs in electron ring
Momentum range is increased by optimizing the phase and strengths of the –I sextupoles.
ex = 19.5 nm
ex = 15.5 nm
np+p/2 from sextupoles to IP
phase adjusted by Dm (x/y)
Dm = 0/6.1°
Dm = -1.1°/7.9°

24. JLEIC Collaboration Meeting Spring 2016
Conclusions
Several types of FF chromaticity correction schemes have been studied and compared for both rings: compact CCB, local non-interleaved –I sextupole pairs, and distributed interleaved –I pairs.
The compact CCB and the non-interleaved –I sextupole pairs provide better chromaticity compensation in both rings with an adequate momentum range and reasonable sextupole strengths.
The non-interleaved –I pairs provide a better dynamic aperture of the ion ring without machine errors due to better cancellation of the sextupole geometric non-linear effects. Effects of magnet non-linear field errors, however, make the aperture comparable to CCB scheme.
Dynamic aperture of the electron ring with the chromaticity correction schemes has been recently estimated  talk by Fanglei Lin.
Next step for the electron ring  replace the thin lens phase trombones with actual quadrupole adjustment and verify the chromatic correction performance.
Other potential improvements  optimize sextupole phase in the ion ring, optimize tune in the electron ring.