SCU Magnet Modelling: Tolerances and Beam Trajectories

1. SCU Magnet Modelling: Tolerances and Beam Trajectories Ben Shepherd Superconducting Undulator Workshop RAL, 28-29 April 2014

2. Effect of undulator errors • Diamond’s undulator specification demands a maximum rms phase error of 3° • To achieve this, we need to tightly control the manufacture of the former and the winding of the coils • What are the effect of small mechanical errors on the phase error?

3. Modelling Errors • A full 3D non-symmetric magnet model (including errors) can give us a beam trajectory, and hence phase errors • BUT this is very time-consuming • There must be a quicker way • Essential if we want to look at many undulators with many different types of errors

4. Recipe for Modelling Errors • Construct ‘perfect’ undulator field map(in Radia or Opera): Bp (z) • Construct undulator with one error: Berr (z) • Subtract field distribution to give error signature δB (z) • Fit a function to the error signature • Generate set of random errors • Convert to field errors (assuming linear) • Add synthesized random errorsto ‘perfect’ undulator field map • Calculate trajectory and phase error φ rms

5. Types of errors considered

6. An example: groove width errorCalculations for one error where  is approximately 4.2mm and z0 is the groove position • The groove width is varied by changing • the width of the two adjacent poles • the coil width • the coil current density • The change in field on-axis is a double-peaked function, similar to the derivative of a normal distribution. m = -0.025 T/mm² Linear for small errors

7. An example: groove width errorCalculations for one undulator Assume normally-distributed ( = 10µm) Field errors for first few periods Start with a perfect sinusoid Add randomly-generated errors for each groove Phase error calculation Fit (z) to straight line Evaluate differences  at poles, calculate RMS Phase error over whole undulator

8. An example: groove width errorCalculations for many undulators Dependence of phase error on width of error distribution There’s a lot of random variation, hence big error bars! Distribution of RMS phase errors for 100 undulators ( = 10µm) Mean: 0.7° This represents about 1200 128-period undulators with random errors on each pole, and was calculated in a few minutes. Full 3D models would have taken significantly longer.

9. Summary of tolerances

10. Checking against Opera-2D model • Built a 2D model with errors to cross-check • Fairly quick to run 16 periods, but sloooow for 128 • However, rms phase error scales well with • Acceptable tolerances seem to be larger • Pole height: ±25µm for 1°(cf ±10µm using numerical method)

11. Design of Undulator Ends • Need to terminate undulator correctly to get trajectory • Straight • On-axis • Also should be OK at 65% current(secondary operating point) • Several constraints: • Poles must protrude above (or be flush with) coils • Pole lengths must be same as main section(except for final half-pole) • Coils must be stacked in whole layers • Odd number of layers makes winding simpler Field  Trajectory

12. Design of Ends Exaggerated vertical scale Gap under coil 0.48 Gap at sides 0.245 Coil layers = ············ 5 7 9 11 1400A (100%) Trajectories (2D model) ······· Pole heights = 2.33 4.55 3.81 4.55 910A (65%)

13. Conclusions • Tolerances • Numerical method to evaluate undulator errors • Produced a table of tolerances • Some are very tight! • Figures are (hopefully) conservative;gives an indication of which dimensions are most important to get right • Ends • 2D model was very useful in producing end design • Design meets constraints, should give good results