July 9

1. July 9

2. Can we get more insight into how the RFQ works? We consider electric quadrupole cross section and the potential function for pure quadrupole geometry (no z dependence). • U and are zero on axis; no axial (z) field is present in this case. • This gives transverse quadrupole fields but no acceleration.

3. Check that the potential function gives the right polarity on the vanes Potential function Horizontal vanes Vertical vanes

4. How to get an axial field? Let’s see what happens if we have an electric quadrupole with unequal apertures? Assume a trial solution for the potential function. There are two unknowns, X and A. What is the solution for X and A in terms of the new geometry?

5. Find potential function that give the correct polaritities on the vanes Our trial solution where X and A are unknowns. We have two equations with two unknowns, X and A. Add and subtract the two equations and we can solve for the unknowns. The results are

6. How to get an axial field for acceleration of particles? • Electric quadrupole cross section with unequal apertures has nonzero potential on axis. • The on-axis potential always has the sign of the electrodes closest to the axis.

7. Getting an axial accelerating field • To produce the axial field suppose we sinusoidally modulate the vanetips along the axial direction. • If this is done with x and y modulations that are 180 degrees out of phase, the on-axis potential will follow the potential variations of the vanetips and a sinusoidal on-axis electric field is produced.

8. RFQ Beam Dynamics Equations

9. Physical interpretation of A: A equals the fraction of the intervane voltage across the unit cell On axis r=0, and Potential difference across the bl/2 cell length is

10. RFQ transit-time factor for the synchronous particle

11. Synchronous acceleration in the RFQ is really the same as for conventional linac structures Interpretation: A is the fraction of the intervane voltage V0 that is applied across a cell of length bl/2. • Note that E0 decreases as b increases.Thus the acceleration becomes less efficient • with increasing b.

12. RFQ transverse focusing • RFQ electric quadrupole focusing is just another example of alternating gradient focusing such as what we have from quadrupole lenses. • The beam particles see focusing with alternating polarity. As the beam sees it, it is just “FD” quadrupole focusing. • As with magnetic lenses, you get net focusing because the beam has a larger displacement in the focusing lens than in the defocusing lens. • What is different is that in the RFQ the focusing alternates in time rather than in space.

13. Transverse Equation of motion Averaging over a cell of length bl/2, assuming x is constant over a cell, we get: quadrupole RF defocusing (when f is negative) This result has the form of the Mathieu equation, also known as the Mathieu-Hill equation.

14. m0 is the mass. Substitute Also known as Mathieu-Hill equation.

15. Transverse equation of motion is the Mathieu-Hill equation (now shown in dimensionless form) w= RF frequency

17. Take average after substituting the trial solution into the equation of motion

18. W/bc is phase advance per unit length, And bl is length (distance synch particle travels) per focusing period.

19. Transverse Equation of Motion in Smooth Approximation(Good approximation for s0<90 deg)

20. Advantages of RFQ focusing • Use of electric rather than magnetic fields is superior for low velocity particles. • Use of RF focusing fields rather than DC fields allows higher peak fields. • The focusing alternates in time but is spatially uniform. When the fields are focusing they focus everywhere in the RFQ. When the fields are defocusing they focus everywhere in the RFQ. • Spatially uniform quadrupole focusing in the RFQ increases the fraction of space used for focusing to 100%. • The short focusing period (bsl) keeps the phase advance per focusing period small which helps keep the beam away from the unstable limit at s=p.

21. Frequency Choice Issues and RFQ Structure Types

22. RFQ Frequency Choice Issues • Lower frequency allows larger aperture, higher beam current limit, lower power density, more relaxed vane alignment tolerances. • Higher frequency gives higher peak surface field limits, lower charge per bunch for a given beam current, and less emittance growth. • RFQ frequency is always constrained by availability of RF tubes. • The frequency choice also determines what type of RFQ structure, 4-vane (above ~200 MHz), or 4-rod (below ~200 MHz). These are the most common RFQ structures.

23. 4-vane and 4-rod are the most common RFQ structures 4-vane 4-rod The choice of RFQ structure affects things like power dissipation, cooling, and ease of tuning.

24. Four-vane RFQ

25. E field region B field region Four-vane cavity overview • Used in the high frequency range above about 200 MHz. • Most common structure for light ions especially protons. • Built with two specially configured end cells to produce a longitudinally uniform fields throughout interior of cavity. • Transverse electric field is localized near vane tips. • Magnetic field is longitudinal localized in four outer quadrants. • Efficiency is high because vane charging currents are uniform along the length of the vanes.

26. Lumped circuit model of RFQ four-vane structure

27. The separation of E and B field regions suggest a description based on a simple lumped circuit model. • Each quadrant is analyzed as a resonant cavity with capacitance C’ and inductance L’. • The four gaps provide separate parallel paths between vanes of opposite potential. Total capacitance per unit length is where the vane length is lv. • We can derive some useful formulas in terms of Cl.

28. Equivalent circuit for the quadrupole mode of a four-vane cavity

29. Determine inductance • Assume B is uniform over outer part of quadrants. Write magnetic flux as where I is total transverse current over vane length lV, and A is effective cross sectional area per quadrant for magnetic field. • Inductance of each quadrant is ratio of flux to current .

30. Formulas from lumped circuit model of four vane resonator • Effective cross sectional area per quadrant as function • of radius of quadrant r: • Resonant frequency • Assuming ejwt time dependence, the peak transverse current on outer wall and B=m0H=m0I/lv in quadrants are

31. RF power Pl dissipated per unit length, stored energy per unit length Wl ,and Q where V is intervane voltage and s is conductivity Wl=ClV2/2

32. Estimating the value of capacitance per unit length Cl First approximation: electrostatic calculation for four rods of circular cross section whose radius of curvature equals aperture radius gives Cl=90 pF/m Better approximation: electromagnetic code SUPERFISH for a four-vane cavity is a weak function of vane radius whose value is about Cl=120pF/m

33. Summary of lumped circuit model of four-vane resonator • The model can be used to estimate properties of a four-vane cavity and show approximate dependence of cavity properties on the parameters. • For accurate calculation of cavity properties for any specific geometry, an electromagnetic-field- solver code should be used.

34. 4-vane RFQ structure characteristics • Vanetip charging currents are distributed uniformly along the vanes so the wall currents see less resistance and less power loss. The 4-vane RFQ is an efficient structure. • Accidental mode degeneracy can occur between the quadrupole mode we want and the dipole modes that we don’t want. • The need to suppress dipole modes may complicate 4-Vane tuning. • The 4-vane structure is generally chosen for RF frequencies above ~200 MHz where this structure is more compact, efficient, and vanes are easier to cool.

35. E field region B field region RF electric and magnetic fields in the central region of the RFQ (away from the ends) These are the E fieldsnear the beam axis From the two-term potential function.

36. But what happens on the ends? End magnetic field lines split and go from a quadrant into the two adjacent ones. > Special end geometry termination makes the central region of the RFQ look like a uniform structure. The termination at the ends creates an LC circuit with the same frequency as the central region > End region provides “open-circuit” boundary conditions maintains continuity of the magnetic flux as the flux turns around at the ends.

37. But there will be distortion of the fields from mode-mixing induced by perturbations and other errors of fabrication • Errors or perturbations induce mixing of fields, especially from other modes nearby in frequency. Fields from these modes can be added as corrections to the unperturbed quadrupole fields. • The amplitudes of these corrections depend on size and location of local frequency errors. Perturbations include RF drive ports, pickup ports, vacuum ports, and other fabrication errors induce these corrections. • Do not think of the perturbations as exciting other modes! Field corrections are induced by errors and perturbations (called mode mixing).

38. Modes in 4-vane RFQ • Azimuthal modes, particularly dipole modes. Unique problem for four-vane resonator. • Higher longitudinal quadrupole modes. Longitudinal modes are a potential problem for any RFQ structure whose length is large compared with the wavelength, including the four-rod RFQ. • Problem modes need to be displaced in frequency from the operating mode by use of appropriate tuners.

39. RFQ dispersion curve example for a 4-vane resonator showing the desired TE210 operating mode at 425 MHz, other longitudinal TE21p modes (p=1, 2, …), and the family of TE11p dipole modes.

40. The are an infinite number of quadrupole modes, each having a different guide wavelength. The ideal quadrupole mode dispersion curve of the four-vane cavity has the classic hyperbolic shape that is characteristic of a uniform structure. wn is the 2p times the frequency of the nth mode w0 is the operating mode frequency , also called the cutoff frequency. lv is the vane length.

41. Dipole modes are a special problem for the 4-vane resonator • If the dipole mode fields are added to the quadrupole mode fields, the beam will see electric deflecting forces. • The lowest frequency dipole mode lies lower in frequency than the quadrupole mode. For certain vane lengths a dipole modes can have nearly the same frequency as the operating mode. (This is known as accidental degeneracy). • “Accidental degeneracy” is a problem because the mixing of modes occurs more readily when modes are close in frequency. • The problem is not that the modes are being excited by the generator but that errors are inducing mixing of mode fields resulting in distortion of fields for the operating mode.

42. Dipole mode configurations The operating mode will be mostly an admixture of these three modes if they are all close in frequency. The 4-vane RFQ must be designed and tuned to minimize the contributions of the two dipole modes.

43. Recall the TE mode nomenclature. • • Modes are labeled TEmnp where m, n, p corresponds to q, r, z. • • m = number of full period variations in q of the field components. • • n = number of zeros of axial field component Bz in the radial direction in range excluding r=0. • • p = number of half period variations in z of the field components. p=0, 1, 2, … • The 4-vane operating mode TE210 has p=0. Higher longitudinal • modes correspond to p=1, 2, 3, … • Dipole modes are TE11p, p=0, 1, 2, … . • Whenever any of these modes lie close in frequency to the RFQ operating mode, we have to find ways of suppressing them.

44. Several methods have been devised to suppress the effects of unwanted modes. • Vane coupling rings that electrically connect opposite vanes ensuring the same vane potentials. Shifts the dipole frequencies upwards eliminating their effect. • Tuning rods that shift the dipole mode frequencies upwards. The simplest approach is rods attached to the end plates that extend into the midplane of each quadrant. • Adjustable slug tuners in all four quadrants along the outer walls. These also allow us to adjust the longitudinal vane voltage profile and compensate for nearby longitudinal modes.