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 1.Construct ‘perfect’ undulator field map (in Radia or Opera): B p (z) 2.Construct undulator with one error: B err (z) 3.Subtract field distribution to give error signature δB (z) 4.Fit a function to the error signature 5.Generate set of random errors 6.Convert to field errors (assuming linear) 7.Add synthesized random errors to ‘perfect’ undulator field map 8.Calculate trajectory and phase error rms φ

5. Types of errors considered Tolerance Groove position Error in machined position of groove (steel) Width of neighbouring poles is affected Groove width Error in machined width of groove (steel) Width of neighbouring poles is affected Coil width Error in machined width of groove (Isopon) Coil stack is compressed/expanded widthways No effect on neighbouring poles Isopon base thickness Error in machined height of groove (steel or Isopon) Coil stack height is increased/decreased Former alignment (vertical and longitudinal) Individual formers move relative to each other Longitudinal former gaps Introduce a gap at joins between formers (keep period) Reduce height of a single peak Pole height Flatness of former datum face

6. An example: groove width error Calculations for one error 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. where  is approximately 4.2mm and z 0 is the groove position m = -0.025 T/mm² Linear for small errors

7. 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 Assume normally-distributed (  = 10µm) Field errors for first few periods Phase error over whole undulator An example: groove width error Calculations for one undulator

8. An example: groove width error Calculations 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 ToleranceAmount for 1° phase error Groove position Error in machined position of groove (steel) Width of neighbouring poles is affected ± 10 µm Groove width Error in machined width of groove (steel) Width of neighbouring poles is affected ± 40 µm Coil width Error in machined width of groove (Isopon) Coil stack is compressed/expanded widthways No effect on neighbouring poles ± 100 µm Isopon base thickness Error in machined height of groove (steel or Isopon) Coil stack height is increased/decreased ± 40 µm Former alignment (vertical and longitudinal) Individual formers move relative to each other ± 3 µm, 10 µrad Longitudinal former gaps Introduce a gap at joins between formers (keep period) Reduce height of a single peak ± 60 µm Pole height Flatness of former datum face ± 10 µm

10. Checking against Opera-2D model

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 1400A (100%) 910A (65%) Trajectories (2D model)

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