EMMA Horizontal and Vertical Corrector Study David Kelliher ASTEC/CCLRC/RAL 14th April, 2007

Introduction Ability to move magnets perpendicular to the beamline in the horizontal plane allows horizontal corrections to be made. Vertical corrections made using kicker magnets. There will be 2 BPMs per cell, providing both horizontal and vertical displacement measurements. No BPMs will be placed in those long drifts with an RF cavity.

BPMs and vertical kicker location Neil Bliss 3/4/07

MADX ‘Correct’ Module The CORRECT statement makes a complete closed orbit or trajectory correction using the computed values at the BPMs from the Twiss table. There are three corrections modes – MICADO, LSQ, SVD. MICADO is used in this study as it tries to minimise the number of correctors used. The MICADO algorithm solves a system of linear equations Where b is the vector of BPM measurements,  is the correction kick vector and A is the beam response matrix to a set of kicks. The algorithm iteratively minimises the norm of the residual vector r using least squares method. At each iteration it finds the corrector that most effectively lowers r.m.s BPM distortion.

Error simulation Errors in the magnet horizontal (  =50  m) and vertical (  =25  m) position simulated by using the MADX function EALIGN. Random errors with a Gaussian distribution, cut- off point at 2  MADX was run with many instances of such randomly perturbed magnets in order to generate useful statistics.

Error distribution – F magnet

BPM location and Horizontal orbit distortion

Horizontal tune / Horizontal Orbit distortion 1 seed used to simulate random alignment errors

Energy Scan 1 seed DFDF

10 MeV 50 seeds DFDF 1234

15 MeV 50 seeds DFDF 1234

Energy Scan 1 seed DFDF 1234

Energy Scan 1 seed DFDF 1234

Variation of Corrector strengths

Horizontal Correction - Conclusions No optimal position for BPMs can be inferred from this study. Outside the vicinity of energies which correspond to integral tunes, the difference in orbit correction accuracy due to BPM position is of the micron order (if all available correctors used). Position of BPMs down to engineering considerations. Corrector strengths were allowed to vary in this study (not feasible in reality). How to find corrector strengths, constant over energy range, which best reduce horizontal orbit distortion?

Number of correctors and vertical orbit distortion

Vertical Tune / Orbit Distortion 1 seed used to simulate random alignment errors

1 Corrector – Variable Strength

1 Corrector – Constant Strength

2 Correctors – Variable Strength

2 Correctors – Constant Strength

Conclusion Due to strongly varying phase advance per cell over the energy range, it is difficult to correct with constant corrector strength There is no simple way to solve this problem using existing MADX routines. A smart interpolation method should be used to find the best set of correctors to reduce both vertical and horizontal orbit distortion over the energy range.