PHY862 Accelerator Systems Hadron linacs (protons, H-minus, ions) Peter N. Ostroumov Professor of Physics Michigan State University

Content LINAC layout, LINAC systems Proton/H-minus linacs RFQ Pulsed and CW linacs Linacs for light ions (protons, H-minus, deuterons) Linacs for heavy ions up to uranium Proton/H-minus linacs RF system Focusing Accelerating structures RFQ RF resonators Beam dynamics Heavy ion linacs Multi-charge acceleration Literature: https://lib.msu.edu/searchresults/?Ntt=RF+Linear+Accelerators+Wangler P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Conditions for RF acceleration Two conditions for RF acceleration should be fulfilled: The wave has electric field component in the direction of particle motion The wave phase velocity is equal to the particle velocity: synchronism Two periodic systems can be used for the RF acceleration Periodically loaded waveguides: acceleration by a traveling wave. Periodic reflections from the conducting walls reduces phase velocity of the wave below the speed of light So far traveling wave structures are use in electron linacs only Periodic coupled resonator structure: acceleration by a standing wave. Standing wave is composed of a sum of two traveling waves prorogating in opposite directions Room temperature linacs Single resonators structures Mainly superconducting cavities P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Transit time factor Electric field distribution in the accelerating gap in resonator Note: sometimes q is the charge and sometimes q is a number of electrons removed from an atom General expression for the energy gain P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Energy Gain  is the synchronous phase Note : q or qe in ion accelerator Transit time factor ( T or TTF) In linear hadron accelerators the phase is referenced to the crest P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Transit time factor for even function Ez If the electrical center coincides with geometrical center If the velocity change in the gap is small P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Transit time factor in DTL gap, simplified Some approximations g Ez z g/2 -g/2 P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Transit time factor for a multi-gap SC resonators Ez is the odd function Ez is the even function P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Typical RF Linac structure Hadron RF Linac Layout Typical RF Linac structure Front end Medium-energy section High-energy section Drift Tube Linac (DTL) Separated DTL (SDTL) IH-structure SC cavities Ion source Radio Frequency Quadrupole Coupled Cavity Linac (Side coupled structure Disk-and Washer Structure Annular Coupled Structure) SC Cavities (Elliptical Spoke-loaded TEM-class) Frequency jump Lattice transition P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Hadron Linear Accelerator Systems Ion source Accelerating structures Standing wave structures (resonators) To reduce the cost of the linac, radio frequency can change as velocity increases Focusing structure Pulsed or DC power supplies RF power amplifiers Based on vacuum tubes, klystrons, solid state HV modulators, cooling system Beam diagnostics Sensors and electronics Control system Cryogenic system Vacuum system Provide residual pressure below 10-7 Torr In SC resonators, vacuum is ~10-9 Torr P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

RF linacs RF Linacs CW NC* SC Pulsed NC ATLAS ISAC-I (RFQ, IH) ISAC-II RIKEN inj. SARAF RFQ ANL RFQ FRIB RFQ RISP RFQ GANIL RFQ ATLAS ISAC-II INFN ReA3 SARAF ADS-IMP FRIB ADS front end SPIRAL-2 PIP-II EURISOL LANSCE Synchrotron Injectors (FNAL,KEK, CERN, IHEP….) MMF (Moscow) SNS CSNS SNS ESS CSNS CERN SPL *Low-energy, several MeV/u Heavy-ions P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Typical parameters of hadron linacs Normal conducting (usually pulsed machines due to limitations of thermal issues ) Beam current: up to 200 mA Beam energy: up to1 GeV for protons or H-minus Uranium (UNILAC - GSI): up to 11 MeV/u Duty cycle up to 12% (LANSCE) Superconducting ion accelerators (CW) ATLAS 50 SC cavities, ~70 MV total voltage 238U up to 10 MeV/u PIAVE-ALPI (INFN, Legnaro, Italy) 40 SC cavities, ~50 MV total voltage 132Xe up to ~7 MeV/u ISAC (TRIUMF) –about 40 MV total voltage FRIB: 200 MeV/u Uranium, 400 MeV/u light ions, 400 kW beam power, CW SNS (NC up to 187 MeV and SC from 187 MeV to 1 GeV) 1.4 MW proton beam on target Projects FNAL PIP-II: 0.8 GeV, 1 mA CW GANIL: 5 mA, 40 MeV, q/A=1/3 SARAF: 5 mA 40 MeV, q/A=1/2 IFMIF: 125 mA, 40 MeV deuterons ESS: pulsed, 2 GeV, 5 MW MYRRHA, CW, 600 MeV, 4mA P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Standing wave structure: DTL (Alvarez)- drift tube linac Long cylinder resonator Loaded with drift tubes Electromagnetic field is TM010 like Drift tubes are usually used to house focusing devices: magnetic quadrupoles Traveling wave structures are not efficient for low velocities due to high RF losses (heat in the walls) Proton and ion accelerators use standing wave structures Decrease d to reduce 0 P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

DTL Protons, f=198.2 MHz, resonator diameter is ~1 meter From 0.75 to 20 MeV from 90 MeV to 100 MeV P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Drift Tube Linac (DTL) Fall 2018 P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Wideroe or interdigital structure Wideroe or Sloan–Lawrence coaxial-line structure in a π−3π configuration P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

IH and CH structures Focusing triplet Fall 2018 P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

IH and CH structures Very high shunt impedance up to  ~ 0.5 Fall 2018 P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

JPARC H-minus linac Ion Species H- Energy 400 MeV Peak intensity 50 mA* Duty 1.25 % Average beam power 133 kW Cavity type RT DTL, RT CCL Frequency 324/972 MHz Status Under operation P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

J-PARC Linac Layout DTL SDTL 　　 Drift-Tube LInac　　　　　　　 　Separated DTL　　　 Annular-Ring Coupled Structure (ACS) P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

J-PARC DTL F=325 MHz, H-minus, 3 MeV to 50 MeV H-minus accelerators are popular as injectors to synchrotrons P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Side coupled structure, LANSCE, SNS section Energy range: 100- 800 MeV High shunt impedance ~50 M/m See Lecture 7-10, p. 108-110 Room temperature Known as CCL – coupled cavity linac P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Annular coupled structure at JPARC 972 MHz Axially symmetric coupling cell High shunt impedance P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

High power amplifiers for DTL P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Klystron P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

805 MHz Klystron Klystron in its solenoid mounted on its pulse transformer P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

JPARC klystron gallery Klystron gallery of the J-PARC linac 2011.09 972MHz 330m 324MHz P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" P.N. Ostroumov Lecture 11 PHY862 "Aceclerator Systems" Fall 2018

Fixed velocity and variable velocity accelerating structures Normal Conducting Beam β/2 Normal or Super Conducting Beam P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" βOPT/2 P.N. Ostroumov Linac Overview - Introduction Fall 2018 28 June 15, 2014

CW Linacs 200 MeV/u FRIB linac Fall 2018 P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

FRIB Linac configuration Cavity Type QWR HWR 0 0.041 0.085 0.285 0.53 f [MHz] 80.5 322 Va [MV] 0.810 1.80 2.09 3.70 Eacc [MV/m] 5.29 5.68 7.89 7.51 Ep /Eacc 5.82 5.89 4.22 3.53 Bp /Eacc [mT/(MV/m)] 10.3 12.1 7.55 8.41 R/Q [Ω] 402 455 224 230 G [Ω] 15.3 22.3 77.9 107 Aperture [m] 0.036 0.040 Leff ≡  [m] 0.153 0.317 0.265 0.493 Lorenz detuning [Hz/(MV/m)2] < 4 Specific QO@VT 1.4E+9 2.0E+9 5.5e+9 9.2E+9 QL 6.3E+6 1.9E+6 5.6E+6 9.7E+6 N= 12 100 72 148 P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

High power heavy ion linac is required for production of radioactive beams Beam power is limited by available current from the ECR ECR LINAC Target St1 St1 St1 Ions Uranium Energy 400 MeV/u Power 400 kW q is the ion charge state P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Stripper effect on beam parameter Stripping lowers intensity in each charge state Main issue is the stripper damage due to (a) heating (b) radiation P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Effective shunt impedance of accelerating structure From Yakovlev’s lecture, p. 45. V is the effective voltage which includes transit time factor, R is the shunt impedance of the cavity In Linacs we also use effective shunt impedance per unit length, L is the length of the accelerating structure The maximum accelerating field in resonators is limited with breakdown field EPEAK Depending on specific type of the resonator, E0 is lower than EPEAK by factor of 2-6 P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Peak fields in accelerating cavities Normal conducting structures made from copper Kilpatrick limit was introduced in 1950s, it is an empirical formula In modern structures, electric field exceeds Kilpatrick limit by a factor of 1.5 – 2.0 Superconducting structures Peak magnetic field is limited by quench, theoretical value is ~200 mT at 2K Peak electric field is limited by the surface quality. ~120 MV/m can be achieved. Operational values are lower Peak fields can not be measured These ratios are known from the simulations of the resonator design EACC can be obtained experimentally from the stored energy P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Linac economics The linac cost is sum of capital and operation cost The capital cost is the cost of accelerating structure The operational cost is the electric bill and maintenance effort The cost of a linac depends from the choice of an average accelerating gradient Total cost, L is the length of linac Capital cost per meter CL Capital cost per watt of power CP Energy gain Power loss in the resonators’ walls, E includes TTF Beam power P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Linac cost Total cost Structure power cost Beam power cost Structure length cost Cost E (MV/m) Accelerating field can be limited by breakdowns in resonators P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Continuous Wave Linac (100% duty cycle): NC or SC ? Required wall plug power to create accelerating field where  is the efficiency of the RF generator Typical example: 1 GeV CW linac Superconducting CW linac is much more economic than NC Both pulsed or CW SC linacs require NC front end for ~0.1 to 10 MeV/u depending on q/A and duty factor Transition energy is higher for pulsed linacs: SNS – 187 MeV ESS - 50 MeV P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Notations Axial distance s z Phase Circular Accelerators Hadron Linear Accelerators Axial distance s z Phase With respect to zero crossing (sin-like) With respect to maximum (cos-like) Particle kinetic energy U W Energy gain U W Energy deviation E W Phase advance of transverse oscillations   (sometimes, ) P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Radial dependence of the accelerating field See Lectures 7-10, p 19,20 The radial dependence of the accelerating field is notable only for hadron accelerating structures due to <c The Bessel functions appear as a result of wave equation solution in axially-symmetric structures P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Longitudinal motion Compare to Lecture 5, now we have radial dependence of the accelerating field In most of accelerators particles perform radial oscillation close to axis in the way that << 1 and the value of the modified Bessel function is close to 1. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Hamiltonian 𝑊 is generalized momentum which is canonically conjugate to the generalized motion coordinate  Hamiltonian describes particle oscillations around synchronous particle. If we assume that particle energy and velocity are changing slowly during particle oscillations then the Hamiltonian does not depend on time and it is a constant of motion. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Phase space trajectories Potential energy Phase trajectory equation for each value of H Separatrix extension in phase is ~3s Relief of potential function and a family of phase trajectories 42 P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Separatrix For stability condition, the synchronous phase must be negative The value of Hamiltonian, corresponding to separatrix, is New variable, more commonly used in Linacs Separatrix equation P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Acceptance . P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Accelerating field and phase trajectories This separatrix is plotted for a conservative approximation (“fish”) These trajectories include acceleration, non-conservative approximation (“golf club” shape) 4. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Stable and unstable phases in standing wave P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" 1 46 Fall 2018

Small longitudinal oscillations Matched Mismatched P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Adiabatic invariant theorem for small oscillations The phase trajectory of small oscillations is an ellipse The amplitude and frequency of oscillations are slowly varying functions of time – apply adiabatic invariant theorem for harmonic oscillations, Sa=const P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Theory of multi-q beam acceleration Energy gain per nucleon q is the ion charge state, A is the mass number Fixed velocity profile (RFQ, RT DTL), energy gain per nucleon will be the same for any q/A if Velocity is defined from energy gain per nucleon Variable velocity profile (SC Linac) E0=const, Tune phases of individual cavities Multi-q heavy-ion beam acceleration P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Synchronous phase as a function of uranium ion charge state Synchronous phase as a function of uranium ion charge state. The designed synchronous phase is –30 for q0 =75. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Synchronous phase of multi-q beam Single accelerating gap E q=75 g (z) -0.15 -0.1 -0.05 0.05 0.1 0.15 Distance, m w t q=77 q=73 Earlier arrival Later arrival S P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Separatrix and small longitudinal oscillations P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Uranium beam stripping and total voltage, 400 kW, 400 MeV/u Bunching efficiency =80% q V, MV Efficiency (%) St. IECR (pA) 29 3790 100 no 5.25 29-82 1677=458+1219 25 1 21 29-71-89 1352=379+973 6.3 2 83 29-59-81-89 1292=319+973 1.33 3 395 Multiple charge state acceleration St. From ECR After St1 After St2 After St3 Efficiency (%) IECR (pA) 2 28-29 69-73 88-91 - 147 3.6 3 58-61 79-83 97 5.4 P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

RF Defocusing in Particle Accelerator Electric field lines between the ends drift tubes. If accelerating, the field is focusing at input and defocusing at output. While field level is increasing while particles cross the gap to provide longitudinal beam bunching, the defocusing effect is larger. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Defocusing due to accelerating field Cylindrical coordinate system Apply Maxwell’s equation, Lectures 7-10, p.12 Vacuum, =1 P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Field Distribution Accelerating gap of an resonator Accelerating field distribution Radial field distribution P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Earnshaw's Theorem Earnshaw's theorem states that a collection of point charges cannot be maintained in a stable stationary equilibrium configuration solely by the electrostatic interaction of the charges. This was first proven by British mathematician Samuel Earnshaw in 1842. It is usually referenced to magnetic fields, but originally applied to electrostatic fields. It applies to the classical inverse-square law forces (electric and gravitational) and also to the magnetic forces of permanent magnets and paramagnetic materials or any combination (but not diamagnetic materials). Effective potential created by static field P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Equations of motion in x - and y - directions (Hill’s Equations) (see S. Lund’s slide 23 and 24): The dissipative term is negligible in Linacs Focusing function for simple FD or FODO structure No accelerating field With the accelerating field Lp is the length of focusing period P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Grid or Foil Focusing of Charged Particles Foil or grid focusing—the defocusing effect is suppressed RF Defocusing effect is suppressed by closing the the drift-tube hole at the exit of the gap with a foil thin enough to be crossed by particles. First test: 1947, Alvarez linac P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Quadrupole Focusing Lorentz Force Arrows indicate direction of Lorentz force acting on positively charge particle moving from the screen. Field is proportional to distance from axis, G- gradient of quadrupole field. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Various Types of Focusing Periods FODO D F D F D F FOD (Doublets) D F F D D F FOF-DOD F/2 D F/2 F/2 D F/2 Triplets Fall 2018 P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems"

Radio Frequency Quadrupoles Short history RF quadrupole focusing properties Creating longitudinal electric fields Electric potential in the RFQ accelerating cell Vane tip shaping Transit time factor Synchronous motion Focusing Potential expansion for arbitrary vane tip shape Examples of beam physics design Multi-harmonic buncher upstream of the RFQ Different resonator structures for RFQs 4-vane structure: advantages and disadvantages P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Quadrupole mass-spectroscopy Different focusing effect for ions with different q/A Can transport selected q/A, all other ions will be lost inside the structure For slow ion beams, frequencies <several MHz P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Stability of different ions in the mass-separator Phase advance as a function of mass-to-charge ratio, A/q for different values of focusing field Ion motion for 0 above 180 is unstable Transfer matrix for periodic structure (Yue Hao’s lecture) If , the transverse motion is unstable and ions are lost P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Radio frequency quadrupole RF voltage is applied to 4 electrodes (vanes) a Potential function in cylindrical coordinates Horizontal vanes Vertical vanes P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Axial accelerating field To produce the axial field suppose we modulate the vane tips 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 vane tips and a sinusoidal on-axis electric field is produced. Longitudinal electric field is between A and B Point B Point B P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Point A Fall 2018

Short history 4 electrodes with RF voltage have been used for mass separation since 1950s I.M. Kapchinski and V.A. Teplyakov invented RFQ. The first publication in 1969 in the Proc. of HEP Accelerator Conference First operational RFQ – 1972, Protvino, Russia Linear Accelerator Structures with Space-Uniform Quadrupole Focusing P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Significant contribution by Los Alamos group P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

CW Radio Frequency Quadrupoles ANL FRIB GANIL RAON IMP f, MHz 60.625 80.5 88.0 81.25 Type 4-vane with windows, brazed 4-vane, brazed 4-vane, bolted A/q 7 3 6.8 V, kV 70 60-115 100-113.5 50 – 138.5 Kp 1.42 (1.6) 1.63 1.65 1.7 1.55 P, kW 60 90 177 94 57 L, m 4 5 6 W, keV/u 300 500 750 P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Vane tips P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Potential distribution The accelerating gap dimensions are much smaller than RF wavelength Electrostatic approximation can be applied Generalized solution can be found as a solution for Laplace equation with quadrupolar boundary conditions transversely and periodic conditions in z Infinite Fourier-Bessel series For “ideal” vane tip electrodes the potential is described as This formula for quasi-electrostatic potential was introduced by I.M. Kapchinskiy P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Electrode surface Fall 2018 P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Physical interpretation of A On axis, r=0 Potential difference across the bsl/2 cell length A equals the fraction of the accelerating voltage P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Transit time factor and energy gain P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Synchronous particle in the RFQ As in a DTL, the synchronous particle “sees” the accelerating field at phase s when the particle is located in the cell center Cell center the cross-section has exact quadrupole symmetry P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Electric field Cylindrical coordinate system Time dependence Fall 2018 P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Transverse focusing This is Mathieu-Hill equation Simplifications P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Apply “smooth” approximation Valid for 0 <90 Typical value of 0 40, in high intensity RFQs, 40 <0 <90 Focusing DOES NOT depend from ion velocity P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

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 About twice of Kilpatrick limit More conservative in CW RFQs, ~1.5 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 defocus 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 (beta lambda) keeps the phase advance per focusing period small which helps keep the beam away from the unstable limit at =π. Very efficient focusing for space charge dominated beams P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Longitudinal motion The same as in DTL Frequency of small longitudinal oscillations P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Beam physics design of RFQs Define vane tip modulation for machining Design the geometry of the RF structure Kapchinski –Teplyakov beam dynamics design Keep bunch physical length constant along the RFQ to minimize space charge effects Los Alamos introduced the beam dynamics design for proton/H-minus RFQs Use radial matcher to match static phase space to dynamic phase space Shaper Gentle buncher Acceleration P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Vane tip modulation S1 S2 S3 S4 S5 S6 S7 S8 S9 x, y z L vanes LRFQ Lf,in Lf,out R0 S1 S2 S3 S4 S5 S6 S7 S8 S9 P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

RFQ for heavy ions Must operate for wide range of q/A – charge to mass ratio Requires different level of RF power in the RFQ resonator Beam space charge is not significant in heavy ion RFQs Requires low frequency of the RFQ resonator: beam velocity after ion source is low Heavy ion RFQs are used in the front end of CW SC linacs: must operate in CW regime Focusing by RFQ To provide we need to increase  (decrease frequency) Operational frequencies of heavy ion RFQs ~60-100 MHz This results to long RFQs P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

RFQ potential for arbitrary shape of vane tips P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

RFQ resonators 4-vane structures are 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. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

2.5 MeV proton RFQ P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

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. IH resonator (-type) 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. P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

6.7-MeV RFQ at LANL P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

4-rod structures Mode mixing is not an issue Suitable for low frequencies 200 MHz Good for pulsed operation Shunt impedance is lower by factor of 3 compared to 4-vane structures These structures have proven in operation Have not seen detailed 3D simulations Some asymmetric distribution especially for magnetic field P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

ReA3 RFQ P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Beam matching in FODO channel If =0, the ellipse is upright: easy to define matching condition to the focusing channel P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Smooth approximation, beam matching, zero current Mismatch beam envelope oscillation, linearize the envelope equation Beam envelope oscillates with twice of betatron frequency x’ Matched ellipse P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Unmatched Beam in Periodic Structure P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Zero current matched beam in periodic focusing channel 0=43 P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Zero current mismatched beam in periodic focusing channel 0=43 One betatron oscillation takes place over 8.4 periods One oscillation of the envelope takes place over 4.2 periods P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

BACKUP SLIDES P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Real linac for ion beam acceleration Pillbox or coaxial /4 (Quarter Wave Resonator) and /2 (Half Wave Resonator) cavities can accelerate particles A string of pillbox (or /4 and /2) cavities powered individually can be tuned to accelerate particles synchronously Continuous wave (100% duty cycle) superconducting accelerators The cavity array can be constructed in one RF structure DTL RFQ Coupled structures Very economic for pulsed (low duty cycle) accelerators P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Advanced EM Optimization of SC resonators Advanced EM optimization : outer conductor: form cylinder to conical shape Drift tubes are highly optimized to reduce EPEAK 2.5 deg drift tube face tilt to compensate beam steering effect 109 MHz Frequency 109.125 72.75 MHz beta 0.14 0.077 U0 at 1 MV/m 0.4 0.39 J bl/2 39 32 cm EPEAK at 1 MV/m 5.0 4.6 MV/m BPEAK at 1 MV/m 92 76 Gs G 40 26 Ohm Rsh/Q 548 575 Voltage per cavity 2.1 2.5 MV Dynamic LHe load at 4K 6 11 W 72.75 MHz P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018

Radial dependence of the accelerating field Traveling waves in the infinite periodic structures Only one harmonic effects on beam acceleration for which 0=s=/k0c Accelerating cavity is an resonator therefore the field is standing wave P.N. Ostroumov Lecture 11 PHY862 "Accelerator Systems" Fall 2018