RF Cavity Design Part 2/3 Combined slides of Erk Jensen (CERN BE/RF) CAS 2013 Mauritzio Vretenar (CERN BE/RF) CAS 2013

Accelerating gap Erk Jensen

Gap of PS cavity (prototype) We need a voltage over gap Gap of PS cavity (prototype)

Drift Tube Linac (DTL) – how it works For slow particles! e.g. protons @ few MeV The drift tube lengths can easily be adapted. electric field Erk Jensen

Drift tube linac – practical implementations Erk Jensen

Accelerating gap We want a voltage across the gap It cannot be DC, since we want the beam tube on ground potential. Use 𝐸 ∙𝑑 𝑠 =− 𝑑 𝐵 𝑑𝑡 ∙𝑑 𝐴 For an empty cavity frequency is determined its diameter Frequency if the lowest accelerating mode TM010 is 𝜔 010 = 𝑐 2.405 a To get small cavity size short wavelengths are used Electron linacs typically λ=10 cm (f=3 GHz) and cavity radius a=3.38cm Storage rings λ=60cm (f=500MHz), a = 23 cm But for small low energy rings lower frequencies are needed Below 1 MHz Photo: PS 19 MHz cavity (prototype, 1966) Erk Jensen Erk Jensen Erk Jensen Erk Jensen

Ferrite Cavities For a pillbox cavity filled with a material frequency for the lowest accelerating mode TM010 is 𝜔 010 = 𝑐 𝜖𝜇 2.405 a Where ε and μ are dielectric and magnetic permeabilities So the frequency can be lowered by using high μ material Material that increases inductance, acts as “open circuit” Materials typically used: ferrites (depending on f-range) magnetic alloys (MA) like Metglas®, Finemet®, Vitrovac®… Resonantly driven with RF (ferrite loaded cavities) or with pulses (induction cell) Price to pay: large power losses due hysteresis and eddy currents, the material must be cooled gap voltage Erk Jensen Erk Jensen Erk Jensen Erk Jensen

Large Frequency Range Ferrite cavities have another advantage for low energy rings: large frequency tuning range Frequency must increase until beam becomes relativistic Applying bias field changes its effective permeability, and hence the frequency Graph: Ferrite B-H loop with added ac field. Incremental permeability μr=Bac/ μoHac varies with operating point.

Simplified model of ferrite cavity

Los Alamos / TRIMF single gap 46 - 61 MHz

CERN PS Booster Cavity 3 – 8.4 MHz RF voltages in anti-phase

CERN Lear Cavity Single gap 0.38 – 3.5 MHz RF voltage on the bias winding is made zero at the bias supply feed point by splitting the winding at the mid voltage point The two windings are crossed and the parallel bias field is in opposite directions in each half of the cavity

CERN PS Booster 1998 0.6 – 1.8 MHz < 10 kV gap NiZn ferrites Erk Jensen

Finemet® Cavity Finemet® is a magnetic alloy exhibiting wideband frequency response and large magnetic field saturation level It allows building wide-band cavities Compared to ferrite cavities, no bias is needed! Example: the CERN PSB 5-gap system: Single Finemet® ring on its cooling plate frequency response: (0.6 … 4) MHz 5 gap prototype … installed in PSB ring 4 Erk Jensen

Finemet RF System MedAustron 6-gap finemet cavity Large instantaneous bandwidth! 0.2 ÷10 MHz, 1 kW solid state amplifier

Characterizing a cavity Erk Jensen

Desired Cavity Properties Delivers high accelerating voltage Vacc Delivers uniform field across the bunch extent Requires minimum RF power Pin Has low power losses Ploss Has large aperture for the beam Does not induce higher order modes (HOMs) Does not develop multipacting Has low break-down rate Is easy and cheap to manufacture Piotr Skowronski

Stored Energy W The electric and magnetic stored energy oscillate in time 90 degrees out of phase. In practice, we can use either the electric or magnetic energy using the peak value. Power dissipation Where Rs is the surface resistance, s is the dc conductivity d is the skin depth

Quality Factor Quality factor describes how much energy can be stored in cavity for a given input power

Acceleration voltage Define 𝑉 𝑎𝑐𝑐 = 𝐸 𝑧 𝑒 j 𝜔 𝛽𝑐 𝑧 d𝑧 . The exponential factor accounts for the variation of the field while particles with velocity 𝛽𝑐 are traversing the gap (see later). With this definition, 𝑉 𝑎𝑐𝑐 is generally complex This becomes important with more than one gap. For the time being we are only interested in 𝑉 𝑎𝑐𝑐 . Attention, different definitions are used! Erk Jensen

Shunt impedance The square of the acceleration voltage is proportional to the power loss 𝑃 𝑙𝑜𝑠𝑠 The proportionality constant defines the quantity “shunt impedance” Attention, also here different definitions are used! Traditionally, the shunt impedance is the quantity to maximize in order to minimize the power required for a given gap voltage. But, too large value may lead to other unwanted effects like easy higher order mode excitation Erk Jensen

R-upon-Q The square of the acceleration voltage is proportional to the stored energy 𝑊 The proportionality constant defines the quantity called R-upon-Q: 𝑅 𝑄 = 𝑉 𝑎𝑐𝑐 2 2 𝜔 0 𝑊 Attention, also here different definitions are used! Erk Jensen

Cavity resonator - equivalent circuit Simplification: single mode IG Beam IB Vacc P Generator Z C L R L=R/(Qw0) : coupling factor C=Q/(Rw0) Cavity R: Shunt impedance : R-upon-Q Erk Jensen

Field rotates by 360° during particle passage. Transit time factor The transit time factor is the ratio of the acceleration voltage to the (non-physical) voltage a particle with infinite velocity would see. The transit time factor of an ideal pillbox cavity (no axial field dependence) of height (gap length) h is: Field rotates by 360° during particle passage. h/l Erk Jensen

Pillbox cavity field (w/o beam tube) h Ø 2a The only non-vanishing field components : Erk Jensen

Reentrant cavity Nose cones increase transit time factor, Round outer shape minimizes losses. Example: KEK photon factory 500 MHz - R probably as good as it gets - this cavity optimized pillbox R/Q: 111 Ω 107.5 Ω Q: 44270 41630 R: 4.9 MΩ 4.47 MΩ nose cone Erk Jensen

Loss factor t f0 Impedance seen by the beam V (induced) Beam IB V (induced) Beam Energy deposited by a single charge q: R/b C L R L=R/(Qw0) C=Q/(Rw0) Cavity Voltage induced by a single charge q: 1 - 1 5 1 1 5 2 t f0 Erk Jensen

Summary: relations Vgap,W, Ploss Gap voltage Vgap R-upon-Q Shunt impedance Energy stored inside the cavity W Power lost in the cavity walls Ploss Q factor Erk Jensen

Beam loading – RF to beam efficiency The beam current “loads” the generator, in the equivalent circuit this appears as a resistance in parallel to the shunt impedance. If the generator is matched to the unloaded cavity, beam loading will cause the accelerating voltage to decrease. The power absorbed by the beam is , the power loss . For high efficiency, beam loading shall be high. The RF to beam efficiency is . Erk Jensen

Beam Loading – allows efficiency choice for CLIC main beam full beam loading – optimum efficiency eta acceleration Erk Jensen

Characterizing cavities Resonance frequency Transit time factor field varies while particle is traversing the gap Shunt impedance Q factor gap voltage – power relation R/Q independent of losses – only geometry! loss factor Circuit definition Linac definition Erk Jensen

Example Pillbox: Erk Jensen

Pillbox: dipole mode electric field magnetic field TM110-mode (only 1/4 shown) electric field magnetic field

Panofsky-Wenzel theorem For particles moving virtually at v=c, the integrated transverse force (kick) can be determined from the transverse variation of the integrated longitudinal force! Pure TE modes: No net transverse force ! Transverse modes are characterized by the transverse impedance in ω-domain the transverse loss factor (kick factor) in t-domain ! W.K.H. Panofsky, W.A. Wenzel: “Some Considerations Concerning the Transverse Deflection of Charged Particles in Radio-Frequency Fields”, RSI 27, 1957] Erk Jensen

CERN/PS 80 MHz cavity (for LHC) inductive (loop) coupling, low self-inductance Erk Jensen

Higher order modes Example shown: 80 MHz cavity PS for LHC. Color-coded: Erk Jensen

Higher order modes ... ... n1 n2 n3 external dampers R1, Q1,w1 IB Erk Jensen

Higher order modes (measured spectrum) without dampers with dampers Erk Jensen

Many gaps Erk Jensen

What do you gain with many gaps? The R/Q of a single gap cavity is limited to some 100 Ω. Now consider to distribute the available power to n identical cavities: each will receive P/n, thus produce an accelerating voltage of . The total accelerating voltage thus increased, equivalent to a total equivalent shunt impedance of . P/n P/n P/n P/n 1 2 3 n Erk Jensen

Standing wave multicell cavity Instead of distributing the power from the amplifier, one might as well couple the cavities, such that the power automatically distributes, or have a cavity with many gaps (e.g. drift tube linac). Coupled cavity accelerating structure (PIMS: PI Mode Structure) The phase relation between gaps is important! Erk Jensen

Coupling accelerating cells 1. Magnetic coupling: open “slots” in regions of high magnetic field  B-field can couple from one cell to the next 2. Electric coupling: enlarge the beam aperture  E-field can couple from one cell to the next How can we couple together a chain of n accelerating cavities ? The effect of the coupling is that the cells no longer resonate independently, but will have common resonances with well defined field patterns.

Drift Tube Linac Maximum coupling, no slots Standing wave linac structure for protons and ions, b=0.1-0.5, f=20-400 MHz Drift tubes are suspended by stems (no net RF current on stem) The 0-mode allows a long enough cell (d=bl) to house focusing quadrupoles inside the drift tubes! E-field B-field

The Side Coupled Linac To operate efficiently in the π/2 mode, the cells that are not excited can be removed from the beam axis → they become coupling cells, as for the Side Coupled Structure.

Example of Side Coupled Structure LIBO (= Linac Booster) A 3 GHz Side Coupled Structure to accelerate protons out of cyclotrons from 62 MeV to 200 MeV Medical application: treatment of tumours (proton therapy) Prototype of Module 1 built at CERN (2000) Collaboration CERN/INFN/TERA Foundation Erk Jensen

H-mode structures Interdigital-H Structure Operates in TE110 mode Transverse E-field “deflected” by adding drift tubes Used for ions, β<0.3 CH Structure operates in TE210, used for protons at β<0.6 High ZT2 but more difficult beam dynamics (no space for quads in drift tubes)

IH Cavity

Multi-gap coupled-cell cavities Between the 2 extreme cases (array of independently phased single-gap cavities / single long chain of coupled cells with lengths matching the particle beta) there can be a large number of variations (number of gaps per cavity, length of the cavity, type of coupling) each optimized for a certain range of energy and type of particle. DTL SCL CH 48 PIMS CCDTL

More examples of cavities Erk Jensen

Examples of cavities PEP II cavity 476 MHz, single cell, 1 MV gap with 150 kW, strong HOM damping, LEP normal-conducting Cu RF cavities, 350 MHz. 5 cell standing wave + spherical cavity for energy storage, 3 MV Erk Jensen

CERN/PS 40 MHz cavity (for LHC) example for capacitive coupling coupling C cavity Erk Jensen

Transverse Deflecting RF Cavities If an electron is travelling in the z direction and we want to kick it in the x direction we can do so with either An electric field directed in x A magnetic field directed in y As we can only get transverse fields on axis with fields that vary with Differential Bessel functions of the 1st kind only modes of type TM1np or TE1np can kick electrons on axis. We call these modes dipole modes 𝐹 =𝑒( 𝐸 + 𝑣 × 𝐵 )

Dipole Modes TE111 TM110 H field B field

TE Modes For TE110 mode transverse kick due to electric field cancels out the kick due to magnetic field Note: for low beta cavities TE modes can be used for deflecting However, when beam pipe is added, the cavity TM110 mode couples to beam pipes TE11 mode The electric and magnetic fields are 90 degrees out of phase in both space and time so that their kicks coherently add.

Applications of RF Deflectors Bunch separation and recombination Bunch length measurements Emittance exchange Crab crossing in colliders

Single cell crab cavity

RFQ

The Radio Frequency Quadrupole (RFQ) At low proton (or ion) energies, space charge defocusing is high and quadrupole focusing is not very effective, cell length becomes small conventional accelerating structures (Drift Tube Linac) are very inefficient use a (relatively) new structure, the Radio Frequency Quadrupole. + − Opposite vanes (180º) − + Adjacent vanes (90º)

RFQ The RFQ uses only electric field to accelerate and focus the beam The wave equation can be replaced with the Laplace equation in cylindrical coordinates The general solution with l+n =2p+1 p=0,1,2…,V/2 the electrode potential, I2n is the modified Bessel function of order 2n and k=2p/bl Taking only the low order solution

RFQ The first term is the potential of an electric quadrupole (focusing term); the second, will generate a longitudinal accelerating electric field. Constants A01 and A10 are determined by imposing the voltage in the electrodes Increasing m one gets more acceleration Decreasing a one gets more focusing

RFQ The operation of the RFQ can best be understood by considering a long electric quadrupole with an alternating voltage on it, Particles moving along the z-axis and staying inside the RFQ for several periods of the alternating voltage, would be exposed to an alternating gradient focusing If the tips of the electrodes are not flat but 'modulated' a part of the electric field is 'deviated' into the longitudinal direction and this field can be used to bunch and accelerate particles

Feeding and tuning the cavity The transmission of the power between the generator and the cavity is done Through a coaxial line (short distances, low power < 100 kW) Through a waveguide. Low losses. Can be cooled The connection between the waveguide and the cavity is done with a short coaxial line with virtually no-losses A ceramic window inside the coaxial cable separates the waveguide from the cavity To bring the cavity into the resonance condition, tuning is done using tuning plungers

END of part 2 of 3