1. Nonlinear Field Quality Checks
Frank Zimmermann
LHCCWG
Based on presentation in Chamonix 2003 including
references therein (O. Bruning, S. Fartoukh, M. Hayes,
B. Jeanneret, J.-P. Koutchouk, R. Ostojic, Q. Qin, F. Schmidt, S. Weisz, and others)

2. LHC optics in a nutshell 8 arcs with 27 FODO cells each (23 regular cells, 2x2 cells for dispersion suppressor) phase advance/cell ~ 90 degrees 8 straight sections, 4 of which low-b insertions

3. 3 types of corrector circuits
spool pieces at dipole ends for b3, b4 and b5; powered differently per arc and per beam; total number of circuits 48
lattice correctors for a2, b2, a3, b3, and b4; total number of circuits 168
correction coils for triplet field errors a3, b3, a4, b4, & b6; total number 40
total # of independent correction circuits: 256!

4. #arc correction circuits & #elements/circuit
type
# circuits
elements/circuit
spool piece b3 16
154
spool piece b4 77
spool piece b5
lattice corr b2 32 8
lattice corr a2 24
4 or 2
lattice corr b3d
11 or 12
lattice corr b3f
10 or 9
lattice corr a3 4
lattice corr b4
8 or 13

5. nonlinear optics tolerances
observable
target
tolerance
dynamic aperture
12 s
0.5-1 s
tune spread
0.0
7x10-3
linear chrom. Q’x,y
>0
0<Q’<2
2nd o. chrom. Q’’x,y
-103<Q’’<103
3rd o. chrom. Q’’’x,y
-5x105<Q’’<3x106
geom. det. dQx,y/de
+/-7x103 m-1
chr.-g. det. d2Qx,y/de/dd
+/-7x106 m-1
numbers include considerations on dynamic aperture,
tune footprint, and off-momentum measurements
[O. Bruning, S. Fartoukh, LHC PR 501]

6. general nonlinear measurement scheme
measure change on phase advance or tune resulting from an off-center orbit in a nonlinear field; orbit can be shifted either by closed-orbit bumps or using dispersion
[study by Jean-Pierre Koutchouk for RHIC low-b insertions]
orbit bump yields:
dispersion bump yields:
MAD design:
differences are of the order of 10%!
10% variation in b3 or b5 corrector strengths can change
dynamic aperture by >1s [Q. Qin, S. Weisz, LHC PN 42, 1996]

7. normal sextupole b3
Local correction (arc by arc) to within 50% needed [M. Hayes, LHC PR 522]
Proposed procedure:
pre-set spool pieces to values determined from magnet measurements
adjust Q’ to ~2 units with 2 (pairs of) families of lattice sextupoles
check local correction with local dispersion bumps or groups of 7p bumps across individual arcs, octant by octant
complementarily or additionally, measure off-momentum (chromatic) phase advance

8. local b3 correction: bumps across one arc
scheme by M. Hayes [LHC PR 522] & O. Bruning [LHC PR 473]
orbit with 7p bump across one arc;
peak value of 3 mm
tune difference vs.
quality of correction
DQ~0.01 for 20% mispowering of b3 spool pieces;
→ we can adjust b3 to within 10% (much better than 50% target)

9. local b3 correction: chromatic phase advance
[Chamonix03]
simulated Df for 1s kick for dp/p=10-3 and dp/p=0;
3 cases: (1) no spool piece mispowered,
(2) sextupole circuit KCS45 missing (BPMs 194 to 257),
(3) decapole circuit KCD45 missing
we can detect missing b3 circuits, but not missing b5!

10. example: chromatic phase-advance measurement in the SPS at 26 GeV
measured Df in the SPS from averaging over four 5-10 mm
(2-4 s) kicks for Dp/p=5x10-3 and Dp/p=0 [R Tomas];
large discrepancies remain –
at the level of the expected missing b3 signature for LHC

11. skew sextupole a3
J.-P. Koutchouk [LHC PN 113]: generation of 2nd o. chromaticity
where ,
and ,
tolerance on
this effect
Proposed procedure: likely no correction needed;
check with off-momentum closest tune approach
tolerance to meet:

12. normal octupole b4
global correction to within 30% needed [M. Hayes, LHC PR 522]
Proposed procedure:
leave lattice octupoles switched off
separate tunes (to reduce contribution from a3) & minimize Q’’
verify detuning with amplitude
If Qy’’ and dQx/dey are zeroed, dQx/dex, dQy/dey and Qx’’ are corrected to about 10%
uncor.: DQ~10-3 at 3s, cor.: DQ<10-4 at 3s

13. normal decapole b5
Local correction (arc by arc) to within 50% needed [M. Hayes, LHC PR 522]
Proposed procedure:
first minimize Q’’’ (global)
measure off-momentum detuning with amplitude; without any correction expect DQ~10-3 at 3s and d=10-3 (8x smaller for single arc)
complementarily, measure chromatic phase advance (effect could be difficult to detect)
measure resonant driving terms (option)
varying kicks to obtain “frequency map’’ (option)

14. nonlinear chromaticity
simulated nonlinear chromaticities in the LHC;
optics with all correctors active, and with missing
b3 or b5 spool piece circuit
though fitted 3rd order coefficient changes, also here
identification of missing b5 circuit looks challenging!

15. SPS as testbed for off-momentum studies?
observable
SPS (meas. 06/20/2002
LHC (simul. example)
LHC tolerance
|Q’’|
600
1500
<1000
Q’’’
-1.8x106
-3x105
>5x105
SPS at 26 GeV as nonlinear as the LHC
(3.5 times LHC tolerance in Q’’’)

16. conclusions
huge number of LHC corrector circuits; only spool pieces considered
if effect of corrector important, we can adjust it with beam-based methods
several alternative diagnostics schemes available for each type of error
methods function well in simulations including BPM noise (except b5)
reality may be different (SPS example)
verify as many procedures as possible in the SPS (or PS,…)