RF Cavity DesignPart 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

Drift Tube Linac (DTL) – how it worksFor slow particles! e.g. protons @ few MeVThe 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 • For an empty cavity frequency is determined its diameter • Frequency if the lowest accelerating mode TM010 is • 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 JensenErk JensenErk JensenErk Jensen

Ferrite Cavities• For a pillbox cavity filled with a material frequency for the lowest

accelerating mode TM010 is • 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 duehysteresis and eddy currents, the material must be cooled gap voltage

Erk JensenErk JensenErk JensenErk 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 gap46 - 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

19980.6 – 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:

frequency response: (0.6 … 4) MHz 5 gap prototype … installed in PSB ring 4

Single Finemet® ring on its cooling plate

Erk Jensen

Finemet RF System MedAustron

MHz, 1 kW solid state amplifier

6-gap finemet cavity

Large instantaneous bandwidth!

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

dVHdVEW 2020

22

.2

;1

;2 0

2

s

sloss RdsH

RP

Quality Factor• Quality factor describes how much energy can be

stored in cavity for a given input power

disspatedpower average

energy storedcavity lossP

WQ

cycleper disspatedpower

energy storedcavity 2 Q

• Define . • The exponential factor accounts for the variation of the field while

particles with velocity are traversing the gap (see later).• With this definition, is generally complex• This becomes important with more than one gap. • For the time being we are only interested in .• Attention, different definitions are used!

Acceleration voltage

Erk Jensen

Shunt impedance

The square of the acceleration voltage is proportional to the power 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

loss

acc

P

VR

2

2

Erk Jensen

• The square of the acceleration voltage is proportional to the stored energy The proportionality constant defines the quantity called R-upon-Q:

Attention, also here different definitions are used!

R-upon-Q

Erk Jensen

Cavity resonator - equivalent circuit

R

Cavity

Generator

IG

ZP

C=Q/(Rw0)

Vacc

Beam

IB

L=R/(Qw0)LC

: coupling factor

R: Shunt impedance : R-upon-Q

Simplification: single mode

R

CL

Erk Jensen

Transit time factor

h/l

zE

zeE

zE

VTT

z

zc

z

z

acc

d

d

d

j

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:

ah

ah

TT22

sin 0101 Field rotates by 360°

during particle passage.

Erk Jensen

Pillbox cavity field (w/o beam tube)

Erk Jensen

The only non-vanishing field components :

h

Ø 2a

a

aT01

101

010

J

J1

,

...40483.201

aa

aB

aa

aa

Ez

011

011

0

011

010

01

0

J

J1

J

J1

j1

ac

pillbox01

0

3770

0

ha

aQ

pillbox

12

2 01

ahah

QR

pillbox

)2

(sin

J

4012

0121

301

Reentrant cavity

Example: KEK photon factory 500 MHz - R probably as good as it gets -

this cavity optimizedpillbox

R/Q: 111 Ω 107.5 ΩQ: 44270 41630R: 4.9 MΩ 4.47 MΩ

Nose cones increase transit time factor, Round outer shape minimizes losses.

nose cone

Erk Jensen

Loss factor

0 5 10 15 20

t f0

-1

0

1

qk

V

loss

gap

2

Voltage induced by a single charge q: t

QLe 20

RR/b

Cavity

Beam

C=Q/(Rw0)

V (induced) IB

L=R/(Qw0)LCEnergy deposited by a single charge q: 2qkloss

CW

V

QR

k gaploss 2

142

2

0

Impedance seen by the beam

Erk Jensen

Summary: relations Vgap,W, Ploss

Power lost in the cavity walls

Gap voltage

loss

gapshunt P

VR

2

2

lossP

WQ 0

W

V

QR gap

0

2

2

W

V

QR

k gaploss 42

2

0

Ploss

Vgap

Energy stored inside the cavity

W

R-upon-Q Shunt impedance

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 .

*Re21

Bgap IV

R

VP gap

2

2

G

B

B

gap I

I

IR

V

1

1

Erk Jensen

Beam Loading – allows efficiencyfull beam loading – optimum efficiencychoice for CLIC main beam

0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .20 .0

0 .2

0 .4

0 .6

0 .8

1 .0

B e a m L o a d in g

acce

lera

tion

eta

eta

acceleration

Erk Jensen

Characterizing cavities• Resonance frequency

• Transit time factorfield varies while particle is traversing the gap

• Shunt impedancegap voltage – power relation

• Q factor

• R/Qindependent of losses – only geometry!

• loss factor

lossshuntgap PRV 22

lossPQW 0

CL

W

V

QR gap

0

2

2

CL

10

W

V

QR

k gaploss 42

2

0

Linac definition

lossshuntgap PRV 2

W

V

QR gap

0

2

W

V

QR

k gaploss 44

2

0

Circuit definition

zE

zeE

z

zc

z

d

dj

Erk Jensen

Example Pillbox:

a

cpillbox

010

aha

h

Q

R

pillbox

)2

(sin

J

4012

0121

301

h

a

aQ

pillbox12

2 01 3770

0

4048.201

S/m108.5 7Cu

Erk Jensen

Pillbox: dipole mode

electric field magnetic field

(only 1/4 shown)TM110-mode

Panofsky-Wenzel theorem

||j FFc

For particles moving virtually at v=c, the integrated transverse force (kick) can be determined from the transverse variation of the integrated longitudinal force!

W.K.H. Panofsky, W.A. Wenzel: “Some Considerations Concerning the Transverse Deflection of Charged Particles in Radio-Frequency Fields”, RSI 27, 1957]

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 !

Erk Jensen

CERN/PS 80 MHz cavity (for LHC)

inductive (loop) coupling, low self-inductance

Erk Jensen

Higher order

modesExample shown:80 MHz cavity PS for LHC.

Color-coded:

E

Erk Jensen

Higher order modes

IB

R3, Q3,w3R2, Q2,w2R1, Q1,w1

......

external dampers

n1 n3n2

Erk Jensen

Higher order modes (measured spectrum)

without dampers

with dampers

Erk Jensen

MANY GAPS

Erk Jensen

What do you gain with many gaps?

PnRnP

RnVacc 22

• 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 .

nPR2

nR

1 2 3 n

P/n P/nP/n P/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

How can we couple together a chain of n accelerating cavities ?

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

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 LinacMaximum coupling, no slots

Tuning plungerQuadrupole

lens

Drift tube

Cavity shellPost coupler

Tuning plungerQuadrupole

lens

Drift tube

Cavity shellPost coupler

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!

B-fieldE-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

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

LIBO (= Linac Booster)

Erk Jensen

H-mode structuresInterdigital-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.

48

Tuning plungerQuadrupole

lens

Drift tube

Cavity shellPost coupler

Tuning plungerQuadrupole

lens

Drift tube

Cavity shellPost coupler

Quadrupole

Coupling CellsBridge Coupler

Quadrupole

Coupling CellsBridge Coupler

Quadrupole

Coupling CellsBridge Coupler

SCL

CCDTL PIMSCH

DTL

MORE EXAMPLES OF CAVITIES

Erk Jensen

Examples of cavities

PEP II cavity476 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

cavity

coupling C

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)

TE111 TM110

H field

B field

Dipole Modes

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 tobeam 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)

+

−

RFQThe RFQ uses only electric field to accelerate and focus the beamThe 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=2/l

Taking only the low order solution

011

2

2

2

2

2

UU

rr

Ur

rr

n lnln

n

nn lkznlkrIAnrA

VzrU cos2cos)(2cos

2),,( 2

20

kzkrIArAV

zrU cos)(2cos2

),,( 0102

01

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

.)(11

,)()(

1

2010201

002

2

10

akaIA

aA

mkaIkaIm

mA

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

RFQ

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