1. Sensitivity of HOM Frequency in the ESS Medium Beta Cavity Aaron Farricker

2. Outline Motivation HOMs in the ESS medium beta cavity Sensitivity of single cells to systematic and random errors Impact on cavity mode frequencies Aaron Farricker2

3. Why are HOMs Important at ESS? Aaron Farricker3 ESS is low frequency (704 MHZ) and low charge (174 pC). Wakefield's are therefore small (1.5 V/pC level) so ESS has decided against HOM couplers. However if a HOM lies on a harmonic of the bunch frequency the voltage in the Wakefield grows drastically resulting in lost bunches and increased cryogenic load. This means the geometry has to be well understood and reproduced upon manufacturing to ensure modes do not lie on harmonics of the bunch frequency.

4. HOMs in the ESS medium beta cavity Aaron Farricker4 All HOMs have low R/Q values as expected. They also all lie a significant distance away from any machine harmonic. All modes in the second band are at least 107 MHz away (closest at 1516 MHz) All modes in the third band are at least 11 MHz away (closest at 1750 MHz) Everything looks good with the cavity design but what happens when errors are added to the design? Single cell in red Full cavity in blue Circuit model is fitted Machine lines as dashed lines

5. Sensitivity of HOMs Aaron Farricker5 E FieldH Field Band 1 Band 3 To look at the effects of geometric errors we need a way to characterise the geometry. We use the parameterisation shown to the left. We can see immediately from the field profiles shown as an example that the first band is sensitive to changes in the cell length and also the cell radius The third band is sensitive near t the walls where we can see the high field.

6. Sensitivity to Individual Parameters Aaron Farricker6 Mid Cell End Cell Here the single cells were simulated in HFSS using 15,000 mesh cells. Each parameter was changed individually over a range of +/- 2 mm. The resulting frequency shifts were fond to be linear and the gradients are given in the tables to the left. We note particularly the high sensitivity of the third band to changes in A and L and importantly the relative sign difference between bands

7. What does this tell us? The third band is particularly sensitive in A and L much more so than the accelerating mode. Several parameters have different signs than the accelerating mode hence tuning the accelerating mode may move the cell frequency drastically in an undesirable direction. This could be a particular problem with the third band which is nearest to a machine line. Aaron Farricker7

8. What about random errors? Upon production there will be errors on all parameters some of which will cancel out. To see the kind of frequency spread expected we simulate a cell with uniformly distributed random error and look at the resulting cell frequencies This is done using Poisson SUPERFISH as it is much faster than a 3D code and still gives accurate results Aaron Farricker8

9. Random Errors Aaron Farricker9 500 cavities were simulated for each data point The results were found to be Gaussian and centred on zero In the plot to the left the sigma of the Gaussian is plotted against the width of the applied errors The sensitivity clearly increases with frequency

10. What is the Effect on the Full Structure Simulating a full structure with errors many times is not feasible due to the scale of the problem Instead we utilise a circuit model to predict the resulting shifts in frequency Aaron Farricker10 Kappa is the cell to cell coupling, Omega is the ratio of the frequency and the resonant frequency squared

11. Cont.. By using the distributions calculated previously as our input we are able to calculate the resulting shifts We do this simply by calculating the eigenvalues of he matrix This method was tested for a three cell pillbox which had the radii of one of its cells changed the results are shown below Aaron Farricker11 Results for a 3 cell pillbox at 704 MHz with similar cell to cell coupling to ESS

12. Effects on ESS Cavity Aaron Farricker12 With increased errors the spread in frequency increases significantly Modes which are further from the pi/2 mode have a shift in their mean as well Each data point contains 5000 simulations generated using the Gaussian fits to the random error data.

13. Cont.. Aaron Farricker13 In this the third band the zero mode is gradually getting closer to the machine line with increased error. The error bars represent 1 sigma of the output data

14. Cont.. Random errors appear to cause little effect when kept below the 500 um level For values less than this the machine line lies over 3 sigma away Aaron Farricker14

15. Effects of Systematic Errors and Random Errors Combined Aaron Farricker15 Here we have looked at the effect of systematic errors on A on the third band combined with random errors The systematic errors are 200, 400, 600 um Clearly with a significant systematic error the risk of hitting a machine line is significantly increased.

16. Conclusion Large shifts in HOM frequency can occur even when the accelerating mode is on frequency Random errors alone unless excessive do not appear to be of great concern Largest concern is a systematic error in A that is corrected in the accelerating mode by changing L which would result in large shifts in the third band Aaron Farricker16