1. February 5, 2004 S. A. Pande - CAT-KEK School on SNS1 Radio Frequency Quadrupole (RFQ) S. A. Pande

2. February 5, 2004 S. A. Pande - CAT-KEK School on SNS2 Introduction The first linac was built in 1928 by Widröe ~ 1MHz 25 kV d=  /2 50 keV K + ions

3. February 5, 2004 S. A. Pande - CAT-KEK School on SNS3 The Sloan Lawrence Structure E. O. Lawrence in association with Sloan built an improved version of Widröe’s linac They used an array of 30 DTs excited by a 42 kV, 7 MHz oscillator to accelerate Hg ions to 1.26 MeV. RFQ is also a Sloan-Lawrence kind of accelerator in which the successive accelerating gaps are  /2 apart.

4. February 5, 2004 S. A. Pande - CAT-KEK School on SNS4 The RFQ It was first proposed by I. Kapchinskii and V. Teplyakov from ITEP Moscow for heavy ions. The first RFQ was built and tested at LANL to get 2 MeV protons. Though invented in the last, the RFQ forms the first accelerator in a chain of heavy ion (including proton) accelerators in recent times.

5. February 5, 2004 S. A. Pande - CAT-KEK School on SNS5 Before 80s almost all of the accelerator facilities for protons and heavy ions, invariably used DC accelerators from few 100 keVs to few MeVs as injectors for linear accelerators which in turn formed the main injectors for the bigger circular machines or acted as sources of charged particle beams. The DC accelerators have certain inherent limitations and difficulties associated with handling of high voltages. The beam has to be bunched before injecting into the linac in order to avoid energy spread in the out coming beam and also to avoid the loss of particles.

6. February 5, 2004 S. A. Pande - CAT-KEK School on SNS6 There was another severe problem associated with the focusing of the beams. The defocusing due to space charge is more severe in the low energy beams. The invention of RFQ, the low energy high current accelerator, helped in overcoming all the difficulties we have seen above. The RFQ simultaneously Focuses Bunches and Accelerates the beam This avoided the need for large DC accelerators and avoided the problems to great extent. Almost all of the DC accelerators were later replaced by RFQ after its invention.

7. February 5, 2004 S. A. Pande - CAT-KEK School on SNS7 Principle of Operation As its name suggests, the RFQ provides electric quadrupole focusing with the electric field oscillating at Radio Frequency -1/2Vcos(  t) 1/2Vcos(  t) Four equispaced conducting electrodes with alternating polarity as we move from one electrode to the next forms the electric quadrupole. Voltage  1/2V 0 cos(  t) is applied in quadru- polar symmetry The electric Quadrupole

8. February 5, 2004 S. A. Pande - CAT-KEK School on SNS8 The off axis particles will experience a transverse force which is alternating in time and this transverse force provides ‘Alternating Gradient’ focusing. The advantage of RFQ is that it provides electric focusing for low velocity particles which is stronger than conventional magnetic focusing. A structure with uniform electrodes along its length will have no component of electric field along the axis and thus will not work as an accelerator. To generate an axial electric field component, the quadrupole electrodes are modulated longitudinally. One pair of electrodes is shifted longitudinally wrt the other pair by 180  so that when the distance from the axis of vertical vanes is at its minimum ‘a’, the horizontal vanes will be maximum apart at ‘ma’.

9. February 5, 2004 S. A. Pande - CAT-KEK School on SNS9 Modulation Modulation of electrodes to generate longitudinal field component a ma A´A´ A Cross section through AA´  /2 One unit cell a ma x z Beam axis m  1

10. February 5, 2004 S. A. Pande - CAT-KEK School on SNS10 The axial electric field component is generated due to the potential difference between the point of minimum separation from axis of vertical vanes (or horizontal vanes) and the point of minimum separation from the axis of the horizontal vane (or vertical vane). In RFQ, the field in successive gaps is in opposite direction and therefore when it is accelerating in one cell, it is decelerating in the next. There are two unit cells per structure period. At a given time every alternate cell will have a particle bunch.

11. February 5, 2004 S. A. Pande - CAT-KEK School on SNS11 The general potential function In RFQ the electrodes in the form of ‘rods’ or ‘vanes’ are placed in cavity resonators to prevent the RF fields from radiating. The issues related to the electrodynamics are distinct from those associated with the beam dynamics. The beam dynamics is confined to a region of small radius near axis as compared to the cavity radius which is proportional to the wavelength. Due to the symmetry property the magnetic field is zero on the axis and also for the region r<<.

12. February 5, 2004 S. A. Pande - CAT-KEK School on SNS12 The consequences are- The wave equation in this region can be replaced by Laplace equation The vanes present well defined boundaries with a potential from which we can analytically derive the fields or We can ask for specific fields and then determine the corresponding vane boundaries. Starting with the Laplace equation in cylin. Coordinates Where U(r, ,z) electric field potential.

13. February 5, 2004 S. A. Pande - CAT-KEK School on SNS13 Solving the above equation by the method of separation of variables, we obtain This is the general K-T potential function a doubly infinite terms. K-T considered only the lowest order terms and proposed to construct the electrode shapes that conform to the resulting equipotential surface. Retaining only s=0 from the first and s=0, n=1 terms from the second summation, we have

14. February 5, 2004 S. A. Pande - CAT-KEK School on SNS14 The two term potential Retaining only s=0 from the first and s=0, n=1 terms from the second summation, we have where k=2  /  ;  =velocity of synchronous particle and I is the modified Bessel function. The potential given by this equation is known as ‘Two Term Potential’ and the dynamics in the RFQ is studied with this potential function.

15. February 5, 2004 S. A. Pande - CAT-KEK School on SNS15 By assuming the horizontal and vertical vanes at +V 0 /2 and –V 0 /2 respectively and putting the boundary conditions at the vane tips, we have We define two dimensionless quantities

16. February 5, 2004 S. A. Pande - CAT-KEK School on SNS16 With these two dimensionless quantities, A 0 =XV 0 /2a 2 and A 10 =AV 0 /2, the two term time dependent potential is written as ---------------- --------------- III The first term gives the potential of an electric quadrupole and the second term gives the accelerating potential. The quantities X and A are known as focusing parameter and acceleration parameter respectively. From the defining equations of X and A we can write X = 1 – AI 0 (ka)

17. February 5, 2004 S. A. Pande - CAT-KEK School on SNS17 By rearranging the last equation, we can write XV + AI 0 (ka)V = V This tells us that the inter-vane voltage V is composed of a part required for focusing (XV) and another required for acceleration (AI 0 (ka)) Similarly, if we put m=1 in the last equation, the vanes are unmodulated and the acceleration parameter goes to zero. A = 0 for m = 1 The RFQ will be just a focusing device. As m increases the acceleration parameter increases and the focusing parameter X decreases.

18. February 5, 2004 S. A. Pande - CAT-KEK School on SNS18 The Field Components The field components are derived from the potential function E r = -  U/  r = -V 0 /2[2(X/a 2 )rcos2  -kAI 1 (kr)coskz] E  = -(1/r)  U/  =(XV/a 2 )rsin2  E z = -  U/  z=(kAV/2)I 0 (kr)sinkz I 1 is the modified Bessel function of first order The first term in Er and E  is the quadrupole focusing field The second term of Er is the gap defocusing term which applies a radial defocusing impulse Since I 1 (kr)  kr/2, the radial impulse is proportional to the displacement from the axis.

19. February 5, 2004 S. A. Pande - CAT-KEK School on SNS19 Voltage and energy gain across a unit cell The voltage across a unit cell can be calculated by where we have used Ez as defined earlier and Lc=  /2 The energy gain is given by  W=qeAVTcos  s For RFQ the transit time factor is T=  /4

20. February 5, 2004 S. A. Pande - CAT-KEK School on SNS20 The Vane tip profiles With time dependent voltages on horizontal & vertical electrodes as +V/2sin(  t+  ) and –V/2sin(  t+  ) and expressing the two term potential in cartesian coordinates by substituting x=rcos  and y=rsin , we have U(x,y,z,t)=V 0 /2[X/a 2 (x 2 -y 2 )+AI 0 (kr)coskz] with U=V/2, we have for the geometry of the vane surface 1=X/a 2 (x 2 -y 2 )+AI 0 (kr)coskz Or x 2 -y 2 =a 2 /X(1-AI 0 (kr)coskz)  The transverse cross sections are hyperbolas

21. February 5, 2004 S. A. Pande - CAT-KEK School on SNS21 The ideal vane tip profile The hyperbolic vane tip profiles

22. February 5, 2004 S. A. Pande - CAT-KEK School on SNS22 But for the ease of machining, and also to control the peak surface electric field, the electrode contours deviate from the ideal hyperbolas. A combination of circular arcs and straight lines is used At the cell centre, i.e. at z=  /4 The RFQ has exact quadrupolar symmetry The x and y tips of the electrode are equidistant from the axis (or have radius r 0 ) given by. r 0 2 =a 2 /X r 0 =aX -1/2 This is known as the average radius of the RFQ. The focusing strength of a modulated structure is equivalent to that of an unmodulated structure with radius r 0.

23. February 5, 2004 S. A. Pande - CAT-KEK School on SNS23 The actual vane tip profiles The vertical vane The horizontal vane One quadrant of RFQ r0r0 

24. February 5, 2004 S. A. Pande - CAT-KEK School on SNS24 Characteristics of RFQ Adiabatic Capture and Bunching Ion source provides a DC beam and thus is injected uniformly from -  to  over one period.  W=~0 and  =360  The RFQ can capture almost all the beam injected and bunch it slowly. In the initial part of RFQ there is no acceleration.The longitudinal electric field which is proportional to AV, is slowly increased by increasing m – the modulation parameter. This provides bunching.

25. February 5, 2004 S. A. Pande - CAT-KEK School on SNS25 Characteristics of RFQ (Contd.) Many cells are devoted to this part in an RFQ. This will not be economical in other linac structures. In RFQ, the cells are very short and many cells can be accommodated in a relatively shorter length. Thus RFQ provides adiabatic capture and bunching. The synchronous phase is kept initially at -90  where we have maximum longitudinal focusing and no acceleration (i.e. the synchronous particle will have no acceleration). Once some rough bunching is achieved, the synchronous phase (  s ) and m are slowly increased further to impart energy and the bunch slowly becomes well defined.

26. February 5, 2004 S. A. Pande - CAT-KEK School on SNS26 The complete RFQ The first RFQ was built at LANL. They divided the whole RFQ in 4 parts. 1. Radial Matching Section (RMS) 2. Shaper (Sh) 3. Gentle Buncher (GB)and 4. Accelerator (Acc) 1. Radial Matching Section (RMS) Matches the input DC beam to the strong transverse focusing structure of the RF quadrupole. In this section m=1, no E z no acceleration, few cells ~5.

27. February 5, 2004 S. A. Pande - CAT-KEK School on SNS27 2. Shaper (Sh) This is a short section which starts the bunching process. This section smoothly joins the RMS where A=0 and  s =-90  to the gentle buncher where A>0 and  s >-90 . This initiates the bunching process. 3. Gentle Buncher (GB) The GB adiabatically bunches the beam and also slowly accelerates to some intermediate energy. Being adiabatic, it forms the major part of the RFQ structure.  s and m are increased ultimately to match those in the accelerator part. 4. Accelerator (Acc) In this part the major emphasis is on the acceleration at a faster rate.  s and m reach their ultimate values.  s ~ -30  and m ~ 1.5 – 2.5.

28. February 5, 2004 S. A. Pande - CAT-KEK School on SNS28 RMSSHAPER

29. February 5, 2004 S. A. Pande - CAT-KEK School on SNS29 Longitudinal profile of the vane tip in 4.5 MeV 50 mA RFQ Accelerator

30. February 5, 2004 S. A. Pande - CAT-KEK School on SNS30 The RFQ Cavity or Resonator Whatever we discussed was the story in the vicinity of the axis where the beam passes through. Let us see how we can generate these fields electro- magnetically. Two types of structures are most commonly used 1. The four rod structure and 2. The four vane structure 3. Split Co-axial cavity is used at few places for heavy ion acceleration. We will study the first two.

31. February 5, 2004 S. A. Pande - CAT-KEK School on SNS31 The four vane structure TE21 mode in circular cylindrical waveguide We introduce the vanes The quadrupole field concentrates near the vane tips Vanes divide the waveguide in 4 quadrants

32. February 5, 2004 S. A. Pande - CAT-KEK School on SNS32 On Quadrant of the RFQ showing electric field lines of quadrupole mode

33. February 5, 2004 S. A. Pande - CAT-KEK School on SNS33 The vanes concentrate the electric field near the axis providing strong quadrupole focusing field Magnetic field which is longitudinal is localized in outer part of the quadrant The vane to vane capacitance reduces the cutoff frequency of the waveguide or the resonant frequency of the cavity. To compensate this the waveguide diameter can be reduced The four vane cavity is obtained by shorting the two ends by conducting plates The boundary condition on each conducting end plate is E tangential =0 This shows a true TE210 mode cannot exist in cylindrical cavity with metallic end walls. Instead the mode will be TE211

34. February 5, 2004 S. A. Pande - CAT-KEK School on SNS34 TE211 and TE210 Modes Resulting Field due to TE211 mode the last subscript denotes the no. of half wavelength variations in z direction Desired Field due to TE210 mode zero last subscript denotes that there is no variation in the longitudinal direction. E Z

35. February 5, 2004 S. A. Pande - CAT-KEK School on SNS35 Therefore gaps are provided between the end wall and the vane ends This produces longitudinally uniform field throughout the interior of the cavity E transverse is localized near the vane tips H longitudinal is localized in outer part of the quadrants vane VANEVANE Side view Top view Cross section through RFQ at an end

36. February 5, 2004 S. A. Pande - CAT-KEK School on SNS36 Eigen modes of a 4 vane cavity There is one more important mode in the 4 vane cavity slightly below in frequency of the quad mode. This is the dipole mode denoted by TE11n The field pattern for quad and dipole modes are shown below x xx x QuadrupoleDipole-1Dipole-2

37. February 5, 2004 S. A. Pande - CAT-KEK School on SNS37 Dipole modes are degenerate modes When a dipole mode is excited, a small potential difference appears across the the opposite vanes where as for the quad mode the opposite modes are exactly at the same potential. If these modes are close to the quadrupole mode, the transverse as well as longitudinal field will be perturbed and the performance will be affected. Therefore the dipole modes should be tuned away from the quadrupole mode. It may happen that the frequency of a higher order dipole mode may fall very close to the quad mode.

38. February 5, 2004 S. A. Pande - CAT-KEK School on SNS38 The longitudinal mode spectrum of a 4 vane RFQ cavity

39. February 5, 2004 S. A. Pande - CAT-KEK School on SNS39 The field stabilization The perturbation caused due to the dipole or othe modes can result in unflat field distribution along the RFQ structure. Due to highly sensitive nature of the RFQ cavity, the machining and tuning errors can also result in dipole mode excitation. Many proposals have been made at many places. Most successful are the ‘Vane Coupling Rings’ introduced at LBNL and Pi mode Stabilizing Loops (PISL) proposed at KEK.

40. February 5, 2004 S. A. Pande - CAT-KEK School on SNS40 Vane Coupling Rings (VCR) The opposite vanes are shorted together forcing them to the same potential. The dipole modes are shifted away. 3 pairs of VCRs are used in structures of 1- 2 m in length Difficult to mount and cooling is a problem

41. February 5, 2004 S. A. Pande - CAT-KEK School on SNS41 PISL Principle The total magnetic flux normal to the surface surrounded by a closed conducting loop is zero. The dipole mode fields will be perturbed more and thus shifted away. Dipole mode x Quad mode

42. February 5, 2004 S. A. Pande - CAT-KEK School on SNS42