1. Fine-Tuning the RFQ End Region

2. “…The Devil is in the Detail” RFQ bulk design very close to completion But before drafting need to check: Repeatability & agreement of codes/meshes Frequency of full 4m RFQ If asymmetry is caused if pumps only on top Tunability of cavity using slug tuners (Saad) Field flatness If small machining details have an effect

3. ANSYS Mesh Quality In ANSYS, results converge for vane-tip mesh < 2mm and quadrant mesh < 15mm. Full 4m RFQ solution needs careful allocation of mesh due to size of problem! Now Compare with other codes…

5. ANSYS Maximum Resolution: 324.131 MHz Superfish Maximum Resolution: 324.137 MHz CST Maximum Resolution: 324.129 MHz

6. Vacuum Port Placement Vacuum ports top & bottom Vacuum ports top only Removing the bottom vacuum port increases frequencies by 25 kHz Open squares indicate theoretical modes, missing due to symmetry, but confirmed real when solving one entire 4m long quadrant

7. Confirming Missing Odd Modes Two quadrants, longitudinal symmetry One quadrant, full 4m long

8. Electric Field in Vane Gap for Different Longitudinal Modes TE210: 324.5MHz

9. Electric Field in Vane Gap for Different Longitudinal Modes TE211: 327.7MHz

10. Electric Field in Vane Gap for Different Longitudinal Modes TE212: 334.6MHz

11. Electric Field in Vane Gap for Different Longitudinal Modes TE213: 345.3MHz

12. Electric Field in Vane Gap for Different Longitudinal Modes TE214: 359.9MHz

13. Electric Field in Vane Gap for Different Longitudinal Modes TE215: 378.0MHz

14. Electric Field in Vane Gap for Different Longitudinal Modes TE216: 397.2MHz

15. Absolute Electric Field of First Four Longitudinal Modes 50% Field drop at ends is unacceptable and cannot be tuned out!

16. Example of a frequency error at a single point x 0 Suppose the local error is a delta function at some point x0. Local error magnitude is defined as  This is the new resonant frequency of the cavity in terms of local frequency error  This relates the cavity frequency change to . is the new wavefunction [Ref: Thomas Wangler, Michigan State University Linac Seminar Series – “RFQ Basics”]

17. Fractional vane-voltage error An analytic solution exists for this summation. It is: Each of the higher modes m contributes a term proportional to the voltage value of each mode at the point of the perturbing error, divided by the mode index m squared so nearest modes in frequency contribute most. [Ref: Thomas Wangler, Michigan State University Linac Seminar Series – “RFQ Basics”]

18. Dependence of the fractional voltage error at each point x on the parameters. The fractional voltage error at each point increases with the fractional cavity frequency error and as the square of the vane length to wavelength ratio. This next graph shows that if the local error at some point x0 causes the local resonant frequency to increase, the local voltage decreases, and vice versa. [Ref: Thomas Wangler, Michigan State University Linac Seminar Series – “RFQ Basics”]

19. V 0 (x) m=0 m=0 and 1 m=1 to 20 Perturbed voltage distribution for problem with a  -function error at the vane end, where x 0 / l V = 0, l V / = 2 and  0 /  0 = 0.01. [Ref: Thomas Wangler, Michigan State University Linac Seminar Series – “RFQ Basics”]

20. Having matcher on/off does indeed drastically affect field flatness due to its local frequency error Matcher On Matcher Off

21. Matcher Off/On Adding the matcher hugely affects the capacitance (inductance to a lesser extent) Want 324MHz (almost midway between these two) which suggests: 1)Increase capacitance as much as possible by reducing material removed for matcher 2)Increase inductance by reducing vacuum volume removed to ensure 7mm end gap This region removed lowers inductance Addition of matcher lowers capacitance Initial design of radial matcher was a circle, tangent to vane at two points. Quadrupole frequency of this end region = 375.125 MHz No radial matcher on cold model: vane had square ends. Quadrupole frequency of this end region = 291.078 MHz

22. H-field Modified Radial Matcher Design 7mm R 5 R 21.8 E-field 334.708 MHz To fine-tune toward 324 MHz, modify cutback radius H-field

23. Fine-tune Inductance by Varying Cut-back radius Original cutback radius = 15mm

24. Rounding Off Corners High magnetic field region, so will affect inductance & frequency To reduce sparking and hot-spots, and to ease machining, the corners will be radiused

25. Fine-tune Inductance by Varying Cut-back radius

26. ANSYS Maximum Resolution: 324.131 MHz Superfish Maximum Resolution: 324.137 MHz CST Maximum Resolution: 324.129 MHz 44.0mm Quadrant radius is uncomfortably close to 324 MHz ∴ Relax it slightly by increasing to 44.1mm 44.0mm

27. Varying Quadrant Radius in CST

28. Fine-tune Inductance by Varying Cut-back radius As a bonus, these changes also improve the Q by ~20%

29. Final Design

30. CST and ANSYS results for final geometry, 4m RFQ with flush tuners. Taking into account meshing accuracy for these large models (see slides 3 & 4), both agree with the frequency being (323.5 ± 0.5) MHz

31. Original Matcher No Matcher Optimised Matcher New matcher design achieves considerably flatter field and ensures (323.5 ± 0.5) MHz along the entire RFQ

32. Questions?